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b/research/selective-borrowing-outer-retune/NOTICE-PR2220-OUTER @@ -0,0 +1,8 @@ +New research prepared by Rohan Arun with substantial OpenAI Codex assistance. Apache-2.0. Parent words and build recipe method derive from Jeremy Kothe PR1274 and Rohan Arun PR1218, with lineage credited in README.md. All inherited mathematical qualifications are retained. +Additional pivot555 and567 composition by Rohan Arun with substantial OpenAI Codex assistance. + +T74 exact-fill owner construction by Rohan Arun with substantial OpenAI Codex assistance. + +PR1377 grid refinement credited to Jeremy Kothe. Outer witness refinement by Rohan Arun with OpenAI Codex assistance; construction unchanged. + +Pivot-566 kernel, words, edits, rebuild.py and rebind.py: PR1625 (mdumitrean). Endpoint/outer-witness stacking, fine_cache.py and oracle stage by jeremykothe2 with oh-my-pi assistance. diff --git a/research/selective-borrowing-outer-retune/README.md b/research/selective-borrowing-outer-retune/README.md new file mode 100644 index 0000000000..16eb2112f0 --- /dev/null +++ b/research/selective-borrowing-outer-retune/README.md @@ -0,0 +1,61 @@ +# Outer retune of PR2260: the PR1673 capped phase-cell fusion outer on the selective-borrowing supplier + +**Conditional final kappa = 0.001683412588542607462330319740399936960503709722451188791314997850799491532148046589685481051097659521487452805635296550068056362547986740649968328627107267620787900814261929971396509432592628352499419999999999999999999998999999999999999999999999** + +That is C − 10^-246 − 10^-222, where C is PR2260's complex supplier endpoint certified here to 10^-200. It is about +0.1733% over PR2260's 3361/2000000 = 0.0016805 and about +0.2177% over PR2220's 0.00167975495612554… + +**What is new here: nothing in the construction.** This package is an *outer retune*. PR2260 (Dugongue) built the finite complex supplier: the 44-alias selective borrowing on PR2226's architecture, 436 helper roles, and a new physical owner packing of the T8 controller. PR2260 assembled it with the retained balanced-prefix outer, whose final kappa is about b/(1+b). PR2220 instead applied PR1673's capped phase-cell fusion outer to its supplier, with strict slacks retuned to the fixed-grid maximum (PR2180 / PR2161). This package applies that same outer, and the same PR2220 ledger code, unchanged in formulas, to PR2260's supplier. The final kappa is then the supplier endpoint minus a fixed-grid backoff. + +## What is checked (`python3 -B verify.py`, assertions on; numpy + SymPy for the vendored outer evidence, as PR2220) +1. **Manifest.** `MANIFEST.json` pins every file. +2. **PR2260 supplier, replayed from scratch.** The package is vendored verbatim at its head `fde6b107` (`vendor/pr2260-selective-borrowing-kappa`, its own manifest pinned). The replay runs PR2260's own steps: native local words, owners, the independent family-factor audit, the finite invoice, and its exact arithmetic including its own 47-row outer at 3361/2000000. + - *One stated deviation.* PR2260's independent receipt pins the SHA-256 of a gzip byte stream, and those bytes depend on the zlib build: CPython 3.9–3.14 here produce different bytes from PR2260's record. Every other receipt key is compared verbatim with PR2260's EXPECTED.json. The *uncompressed* family-factor payload hash in `SUPPLIER-BINDING.json` is a reproducibility pin taken from this package's own replays, identical under CPython 3.11, 3.12 and 3.14. PR2260 did not publish it, and PR2260's recorded compressed hash could not be reproduced on any zlib build available here. +3. **Binding.** PR2260's recounted global profile `{1:176252, 2:30096, 3:30908, 4:10152, 5:4493, 6:18, 7:494, 8:576, 32:11, 39:7, 48:2493}` must equal `SPEC.json`. The other bound inputs are m = 73, W = 7168, rank mass 520704 and deficit 2560, plus the retained bit endpoint 0.001733427039589844234749. +4. **Endpoint** (`certify_endpoint.py`, exact rationals with directed rounding). The moment is Σ_r r H_r/(mW) (m/r)^b. + - Both moments are below 1 at C: the native moment, and the moment with PR2260's retained conservative bad-residue surcharge. + - The conservative moment exceeds 1 at C + 10^-200. + - C = 0.0016834125885426074623303197… PR2260 itself certified only 8417/5000000. +5. **Outer** (`outer/`). These are PR2220's `check_ledger.py` and `independent_check.py`, regenerated byte-identically, together with PR2220's supplier-independent schedule, structure, gauge-free and fine-padding checks and PR1673's 19 exact evidence replays. + - 54 strict rows: PR2220's 63 rows, minus its 11 rows that compare with historical T74 kappas, plus `improvement_over_pr2220` and `improvement_over_pr2260`. + - 17 rejecting controls fire. + - The fixed-grid maximum kappa = C − 1/F − B is asserted. + - *Stale T74 numbers in `outer/*.md`.* Those files are byte-identical to PR2220 (except the pointer note prepended to `outer/GAUGEFREE-INTERIORS.md` and this package's own `outer/ACCRETIVE-SCALE.md`, see below) and quote its T74 instance (DIG = 60000, r0 = 1e-60080, a 2^-199767 grid, 62/63 strict rows, q = 0.0016797549…). The values used here are DIG = 200, R0 = 1e-280 (ZETA = 1e-282, STOP = 1e-286), F = 10^246, B = 1e-222, enclosure grid 2^-1112 and 54 rows (`outer/check_ledger.py`, which derives them from `SPEC.json`; the ledger run prints the 54 rows). +6. **Bundled proofs and the accretive scale qualification.** + - `vendor/pr1786/FINE-PRIME-PADDING.md` (SHA-256 `47d1b1b4df28aadda3cd6bdeb1fbc47290595e9208cd09c1a1f338520d33fdce`) and its retained notice `vendor/pr1786/NOTICE.md` (`dcc1adb2461aa93ca62b6a912dbd8b215599e6af8053dfbacb7423b544dd659a`) are vendored verbatim from PR2220 `fc09a70baf7d6bcff935df226f6fc9e6c6c0131d:research/capped-fusion-t74-k554/vendor/pr1786/`; the proof is byte-identical to PR1786 `c52345caa318fc6b1b6e2659c57d746317a72823:research/fine-prime-padding/FINE-PRIME-PADDING.md`. It is the proof cited by `outer/FINE-PADDING-COMPOSITION.md`; the notice's `outer/check_fine_padding.py` is byte-identical to PR2220's. + - GAUGEFREE-INTERIORS Theorem 4.1 is false as a uniform bound in Lambda, mu (maintainer review of PR1758, https://github.com/CrocSwap/integer-mult-bounds/pull/1758#issuecomment-6203527654). This package relies on the corrected form in `outer/ACCRETIVE-SCALE.md`: log(2 + Lambda + 1/mu) in general, and the original ratio form for unit-diagonal kernels (Lambda >= 1, mu <= 1), which every applied kernel G'_s is, including truncated/rounded kernels and descendants. `outer/check_accretive_scale.py` (exact, run by `outer/run_outer.py`) checks the counterexample, the inequalities and the unit diagonal of the applied kernels. No ledger number or kappa changes. + +## Why the outer is supplier-generic +PR1673's theorem (`vendor/pr1673/capped-phase-fusion/phase_fusion_note.tex`, Thm "Conditional capped-fusion transfer" and §"Updated T74 supplier") is "parameterized by the effective supplier rate". PR1673 was first stated on the T8 selected supplier (b_* = 0.001675462676808…), which is the controller class of PR2260's 72-call T8 supplier; T74 was a later substitution. + +**Checked facts, not inferences:** +- *Whole-call row hosting.* PR2260's arithmetic recount (`vendor/pr2260-selective-borrowing-kappa/arithmetic/verify.py`) requires exactly 72 × 436 owner rows, one per (call, helper). Each row has one (owner_replica, child, port) bin and contiguous axes `range(offset, offset+width)`. Roles are therefore whole; "multi-bin repacking" describes the packing solver, not split residuals. +- *No hidden supplier rate in the outer evidence.* `struct_check.py`, `sched_test.py`, `sched_review.py`, `check_gaugefree.py` and `check_fine_padding.py` contain no supplier rate or kappa literal; their checks are structural. +- *The gain comes from the outer, not a looser claim.* PR2260's own `above_balance` control tests the balanced-prefix outer's margin eps·q ≈ b/(1+b) ≈ 0.0016805 + ~8e-8; 3361/2000000 is the floor of that margin on PR2260's 1e-7 grid (about 8e-8 below the margin), so the gain comes from the different outer, not from PR2260 having claimed less. + +- **Original setting.** It was first stated for the T8 selected supplier (b_* = 0.001675462676808…) and then substituted with T74. The substitution needs only: + - a fixed native dimension and maximum child rank (here m = 73 and r_max = 48, with 73² > 2·48²); + - restoration of arbitrary dirty helper and spectator coordinates, together with the same normalized global transform at each completed-call boundary. PR2260 checks both native orientations under arbitrary dirty inputs; + - the retained interfaces (common-frame / mutable-source realization, whole-call row hosting, semantic precision guard with C_1 = 1, fixed-tape compiler). PR2260 states it uses the same retained interfaces, with C_1 = 1. +- **What a larger stock changes.** A larger finite stock W, PR2260's fresh C_0 < 2^210000 and its row coefficient 16467 < 20161 change fixed constants, not powers of p. +- **Ordering.** The ordering q < a_eff < b < a_* also holds here. + +**Items of PR1673's own T74 substitution review (`vendor/pr1673/.../INDEPENDENT-SUBSTITUTION-REVIEW.md`), mapped to PR2260.** Items marked *checked* were read against the sources by an independent read-only audit; items marked *inherited* are taken from PR2260's own package (`PROOF.md`, `PROOF-LOCAL.md`) and not re-derived here. +- *Product-row stock Q_rows = W_complex^D · W_bit^D.* Checked: W = 7168 changes only a constant of the stock, not a power of p. +- *Semantic-guard / prefix-allowance constant.* T74 used 2048 bits. For PR2260 the guard is its magnitude guard 244λ < 512λ with C_0 < 2^210000. Inherited from PR2260; not re-derived here. +- *Completed global call, no ambient-p address permutations.* PR2260's dirty borrowing is internal to the T8 parent call and is restored at its boundary, so the supplier is used only as a completed global call on complete fields. Inherited from PR2260's PROOF.md / PROOF-LOCAL.md; not re-derived here. +- *436 helpers instead of 480.* Checked: an interface-neutral constant; the owner-row recount is 72 × 436 (see above). +- *Equal-chunk interchange of the bit supplier.* Inherited: the retained bit construction is unchanged from PR2260; not re-derived here. + +## Scope and caveats +- This is a conditional claim, exactly as conditional as PR2260 and PR2220. + - It inherits PR2260's supplier premises: the all-size compiler, physical fixed-tape realization, retained bit construction, exact recovery and analytic transfer. + - It also inherits every premise of PR1673's capped-fusion outer and of the PR1786 / PR1758 / PR2180 refinements bundled in PR2220's outer. +- No new finite construction, no Lean proof and no independent human review are claimed. The retune changes only which already-published outer is applied to an already-published supplier. +- If PR2260's supplier is withdrawn or corrected, this retune falls with it. +- PR2260 is itself a recent PR that has not yet been reviewed, so this result is conditional on it as well. + +## Credits +- **Supplier:** PR2260 (Dugongue), selective 44-alias dirty borrowing and the new physical owner packing. PR2226 (GitHub 1337isnot1337; Cinna Davis), the anticipatory borrowing architecture. Their predecessors are credited in `vendor/pr2260-selective-borrowing-kappa/NOTICE.md`. +- **Outer:** PR1673 (Vlado Vojdanovski, Vlado-V), capped phase-cell fusion, vendored verbatim from PR2220. PR2220 (jeremykothe2; prepared in the Rohan Arun T74 series), the retune method, ledger and outer evidence package. PR2180 / PR2161 (dataisfire), the fixed-grid maximum. PR1786 (chafreaky / Chafik Boukhalfa), fine prime padding. PR1758, the gauge-free interiors. PR1699, the classical matrix boundary. Their cited sources, including Le Gall and ADVWXXZ for rectangular matrix exponents and Trudgian for effective primes, are listed in `outer/*.md`. + +Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0. diff --git a/research/selective-borrowing-outer-retune/SPEC.json b/research/selective-borrowing-outer-retune/SPEC.json new file mode 100644 index 0000000000..6485c2a880 --- /dev/null +++ b/research/selective-borrowing-outer-retune/SPEC.json @@ -0,0 +1,39 @@ +{ + "schema": "outer-retune-spec/1", + "name": "pr2260-selective-borrowing-supplier+pr1673-capped-fusion-outer", + "supplier": { + "pr": 2260, + "head_commit": "fde6b107e26a14da5bac5b66913c914708ea6901", + "package": "vendor/pr2260-selective-borrowing-kappa", + "supplier_kappa": "3361/2000000", + "certified_complex_saving_in_pr2260": "8417/5000000", + "bit_lower": "1733427039589844234749/1000000000000000000000000" + }, + "geometry": { + "dimension": 73, + "calls": 72, + "helpers_per_call": 436 + }, + "expected": { + "global_profile": { + "1": 176252, + "2": 30096, + "3": 30908, + "4": 10152, + "5": 4493, + "6": 18, + "7": 494, + "8": 576, + "32": 11, + "39": 7, + "48": 2493 + }, + "live": 7168, + "rank_mass": 520704, + "deficit": 2560, + "kappa": "3361/2000000" + }, + "outer_refinement": { + "endpoint_digits": 200 + } +} diff --git a/research/selective-borrowing-outer-retune/SUPPLIER-BINDING.json b/research/selective-borrowing-outer-retune/SUPPLIER-BINDING.json new file mode 100644 index 0000000000..e56782176b --- /dev/null +++ b/research/selective-borrowing-outer-retune/SUPPLIER-BINDING.json @@ -0,0 +1,6 @@ +{ + "pr2260_head_commit": "fde6b107e26a14da5bac5b66913c914708ea6901", + "pr2260_manifest_sha256": "a7e8b8b8602ccb42ae5534de8ad5feb0f7c4ac46ba2b8f1ce4cb5aa1a3a33718", + "independent_actual_factors_uncompressed_sha256": "33d9b4a0744997b975e3f426c737062d783bed0c1065df4dac42b8dc173838b4", + "note": "PR2260 pins actual_factors_sha256 of gzip.compress(payload, mtime=0); those bytes depend on the zlib build (CPython 3.9-3.14 on this machine give a06dd8ea.../49c72a31..., PR2260 records 81289c86...). This package pins the uncompressed payload instead; all other receipt keys are compared verbatim." +} \ No newline at end of file diff --git a/research/selective-borrowing-outer-retune/certify_endpoint.py b/research/selective-borrowing-outer-retune/certify_endpoint.py new file mode 100644 index 0000000000..55dc611c01 --- /dev/null +++ b/research/selective-borrowing-outer-retune/certify_endpoint.py @@ -0,0 +1,93 @@ +#!/usr/bin/env python3 +"""Certify the complex supplier endpoint of PR2260's recounted global profile to 10^-DIG. + +C = DIG-digit decimal with conservative_moment(C) < 1 and conservative_moment(C + 10^-DIG) > 1, +and also native_moment(C) < 1. The moment is PR2260's (and PR1673's) sum_r r H_r/(m W) (m/r)^b; the conservative +form adds PR2260's retained bad-residue surcharge 32 m sum(H)/(10^16 W) m^b (vendor/pr2260-.../arithmetic/vendor/moment.py). +Enclosures: exact rationals with directed rounding on a 2^-GB grid (the routines of outer/check_ledger.py, PR1786/PR1673 lineage). +usage: certify_endpoint.py [--write] (default: check the committed ENDPOINT.json) +""" +from fractions import Fraction as Q +from decimal import Decimal, localcontext +from pathlib import Path +import json, sys +sys.set_int_max_str_digits(0) +if not __debug__: raise RuntimeError('Assertions must be enabled') +HERE = Path(__file__).resolve().parent +SPEC = json.loads((HERE / 'SPEC.json').read_text()) +DIG = SPEC['outer_refinement']['endpoint_digits'] +profile = {int(r): n for r, n in SPEC['expected']['global_profile'].items()} +m = SPEC['geometry']['dimension']; W = SPEC['expected']['live'] +assert (m, W) == (73, 7168) and sum(r * n for r, n in profile.items()) == 520704 and m * W - 520704 == 2560 +assert m * m > 2 * max(profile) ** 2 +BAD = Q(32 * m * sum(profile.values()), 10 ** 16 * W) # weight of the conservative rank-1 surcharge term + +GB = int((DIG + 130) * 3.3220) + 16 +ONE = 1 << GB +def _cd(a, b): return -((-a) // b) +def down(x): return Q((x * ONE).__floor__(), ONE) +def up(x): return Q(_cd(x.numerator * ONE, x.denominator), ONE) +LN = GB // 3 + 12; EN = GB // 6 + 12 +def unit_log(y): + z = (y - 1) / (y + 1); p, q = z.numerator, z.denominator; p2, q2 = p * p, q * q + pl = ONE * p // q; ph = _cd(ONE * p, q); sl = sh = 0 + for j in range(LN): + sl += pl // (2 * j + 1); sh += _cd(ph, 2 * j + 1); pl = pl * p2 // q2; ph = _cd(ph * p2, q2) + tail = _cd(2 * ph * q2, (2 * LN + 1) * (q2 - p2)) + return Q(2 * sl, ONE), Q(2 * sh + tail, ONE) +ln2 = unit_log(Q(2)) +def log(x): + k = 0 + while x >= 2: x /= 2; k += 1 + lo, hi = unit_log(x) + return down(lo + k * ln2[0]), up(hi + k * ln2[1]) +def exp(x): + assert 0 <= x < Q(1, 100) + xl = x.numerator * ONE // x.denominator; xu = _cd(x.numerator * ONE, x.denominator) + tl = th = sl = sh = ONE + for j in range(1, EN): + tl = ((tl * xl) >> GB) // j; th = _cd(-((-th * xu) >> GB), j); sl += tl; sh += th + tn = _cd(-((-th * xu) >> GB), EN) + return Q(sl, ONE), Q(sh + 2 * tn + 2, ONE) +LOGS = {r: log(Q(m, r)) for r in set(profile) | {1}} +def moment(b, conservative): + lo = hi = Q(0) + terms = [(r, Q(n * r, m * W)) for r, n in profile.items()] + if conservative: terms.append((1, BAD)) + for r, w in terms: + l, h = LOGS[r] + lo += w * exp(b * l)[0]; hi += w * exp(b * h)[1] + return down(lo), up(hi) + +def decimal_root(): + with localcontext() as c: + c.prec = DIG + 60 + t = [(Decimal(n * r) / Decimal(m * W), (Decimal(m) / Decimal(r)).ln()) for r, n in profile.items()] + t.append((Decimal(BAD.numerator) / Decimal(BAD.denominator), Decimal(m).ln())) + b = Decimal('0.0016834') + for _ in range(60): + f = sum(w * (b * L).exp() for w, L in t) - 1 + d = sum(w * L * (b * L).exp() for w, L in t) + b -= f / d + return int(b.scaleb(DIG).to_integral_value(rounding='ROUND_FLOOR')) + +if '--write' in sys.argv[1:]: + C = Q(decimal_root(), 10 ** DIG) +else: + C = Q(json.loads((HERE / 'ENDPOINT.json').read_text())['complex_lower']) +step = Q(1, 10 ** DIG) +assert (C * 10 ** DIG).denominator == 1 +nat = moment(C, False); con = moment(C, True); rej = moment(C + step, True) +assert nat[1] < 1 and con[1] < 1, 'endpoint not accepted' +assert rej[0] > 1, 'next grid point not rejected (conservative)' +out = dict(schema='pr2260-complex-endpoint-v1', endpoint_digits=DIG, complex_lower=str(C), + native_moment_upper_minus_one=str(nat[1] - 1), conservative_moment_upper_minus_one=str(con[1] - 1), + conservative_moment_lower_minus_one_at_next_grid_point=str(rej[0] - 1), + profile={str(k): v for k, v in sorted(profile.items())}, m=m, W=W, surcharge_weight=str(BAD)) +if '--write' in sys.argv[1:]: + (HERE / 'ENDPOINT.json').write_text(json.dumps(out, indent=1) + '\n') +else: + assert json.loads((HERE / 'ENDPOINT.json').read_text()) == out, 'ENDPOINT.json differs from fresh certification' +with localcontext() as c: + c.prec = 40 + print('PASS_ENDPOINT', 'C =', Decimal(C.numerator) / Decimal(C.denominator), 'DIG', DIG) diff --git a/research/selective-borrowing-outer-retune/outer/ACCRETIVE-SCALE.md b/research/selective-borrowing-outer-retune/outer/ACCRETIVE-SCALE.md new file mode 100644 index 0000000000..8bf0dd647a --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/ACCRETIVE-SCALE.md @@ -0,0 +1,91 @@ +# Scale qualification of the accretive Toeplitz circuit (erratum to GAUGEFREE-INTERIORS Theorem 4.1) + +**Source of the correction.** The maintainer review of PR1758 at +https://github.com/CrocSwap/integer-mult-bounds/pull/1758#issuecomment-6203527654 (retained by this PR, #2265). This note is +this package's own text. The files that carry the uncorrected statement are inherited verbatim and are **not edited**: +`outer/GAUGEFREE-INTERIORS.md` (PR1758 text as bundled by PR2220; only a pointer note is prepended, following PR2220's own +package-note precedent), `outer/FINE-PADDING-COMPOSITION.md` (table row "accretive circuit lemma", obligation 3) and +`vendor/pr1786/FINE-PRIME-PADDING.md` (section 3, "Its accretive shallow-circuit theorem therefore has polylog(p) guard bits"). +Wherever those files cite the accretive circuit theorem, this package relies on the form stated here. + +## 1. The defect + +GAUGEFREE-INTERIORS Theorem 4.1 states, uniformly in Lambda and mu, +`log(max prefix / suffix / inverse-prefix norm) = O(log N log(2 + Lambda/mu) + log N)`. +This is false as a uniform statement: for N = 2 and H = tI (t > 0), Lambda = mu = t, so Lambda/mu = 1, but the completed +circuit has norm t and its inverse norm 1/t; the left side is at least |log t|, unbounded in t. The precision statement +`p + O(log N log(2 + Lambda/mu))` bits has the same defect. (`check_accretive_scale.py` stage A.) + +## 2. Corrected statement + +**Theorem 4.1' (general form).** Let N be a power of two and H an N x N complex Toeplitz matrix with ||H||_2 <= Lambda and +(H + H^*)/2 >= mu I, mu > 0. The circuit of Theorem 4.1 (same rounds, gates, no ancilla) has + + log(max prefix / suffix / inverse-prefix operator norm) = O(log N * log(2 + Lambda + 1/mu) + log N), + +and rounding every coefficient to p + O(log N * log(2 + Lambda + 1/mu)) bits gives operator error 2^-p. +This is the absolute-scale guard already used by the vendored PR1673 `SHALLOW-TOEPLITZ.md` sec. 9 (line 195, +`log(2+M+1/mu)`), on which the proof of Theorem 4.1 rests. + +*Proof changes.* Step (b) of the proof is unchanged and already absolute: every descendant S(a,b) = T_b/T_a satisfies +||S^-1|| <= 1/mu and ||S|| <= Lambda + Lambda^2/mu =: Lambda'; the shears K = A^-1 B, L = C A^-1 have norm <= Lambda/mu. +Step (d) must read "each completed node factor has norm <= poly(N) Lambda'^{O(1)} and inverse norm <= poly(N) mu^{-O(1)}", +hence log B(N) = O(log N log(2 + Lambda' + 1/mu)); and log(2 + Lambda' + 1/mu) = O(log(2 + Lambda + 1/mu)) since +Lambda' <= (Lambda + 1/mu)^3 (Lambda^2/mu <= (Lambda + 1/mu)^3 by the binomial term 3 Lambda^2/mu, and Lambda <= (Lambda + 1/mu)^3 +because Lambda >= mu gives Lambda + 1/mu >= mu + 1/mu >= 2). The only error was replacing the pair (Lambda', 1/mu) by +the single ratio Lambda'/mu. + +**Corollary 4.1'' (ratio form, unit diagonal).** If in addition H has unit diagonal (h_0 = 1), then +Lambda >= |h_0| = 1 (||H|| >= ||H e_0|| >= |h_0|) and mu <= Re = 1, so + + Lambda + 1/mu <= 2 Lambda/mu (2 Lambda/mu - Lambda - 1/mu = (Lambda - 1)/mu + Lambda (1 - mu)/mu >= 0), + +and the bounds of Theorem 4.1 hold as originally written, with log(2 + Lambda/mu) (constant factor absorbed in the O). +The hypothesis is needed **at the root kernel only**: descendants do not keep unit diagonal, but their bounds are in root +quantities, and on unit diagonal Lambda' + 1/mu <= 3 (Lambda/mu)^2, 2 + Lambda' + 1/mu <= 4 (2 + Lambda/mu)^2, +1/mu <= Lambda/mu, shears <= Lambda/mu. So every per-node log factor is <= 2 log(2 + Lambda/mu) + log 4. +(`check_accretive_scale.py` stages B and C: exact polynomial-coefficient checks.) + +*Near-unit fallback.* If |h_0 - 1| <= 1/2 then |h_0| >= 1/2 and Re h_0 <= 3/2, and +Lambda + 1/mu = (Lambda/mu)(mu + 1/Lambda) <= (Re h_0 + 1/|h_0|) Lambda/mu <= (7/2) Lambda/mu (stage D). + +## 3. Truncated and rounded kernels (Corollary 4.2) and boundary pivots (Corollary 4.3) + +* **Corollary 4.2.** Band truncation at W removes only off-diagonal coefficients, and G'_s(0) = e^0 = 1 is stored exactly + (the inherited texts fix it: GAUGEFREE-INTERIORS Section 2, "G'_s(0) = 1 (unit diagonal, as the audited N)"; + FINE-PRIME-PADDING section 3, "rounded Gaussian entries, with diagonal exactly one"). So the truncated/rounded kernel + has diagonal exactly one and Corollary 4.1'' applies to it directly with (Lambda + eps', mu/2), eps' <= mu/2: + Lambda + eps' >= 1 and mu/2 <= 1/2, so the guard is O(log N log(2 + Lambda/mu)). Even if a diagonal were perturbed by + |delta| <= mu/2 <= 1/2, the near-unit fallback gives the same order with constant 7/2. +* **Corollary 4.3.** The pivot bounds ||P_j^-1|| <= 1/mu and ||P_j|| <= Lambda + Lambda^2/mu are absolute statements and + remain true; where they feed a circuit or precision guard, the root kernel G'_s is unit diagonal, so the descendant + inequalities of Section 2 convert them to the ratio form. +* **Descendant / shear bounds** (proof step (b)): unchanged as absolute statements; in ratio form only via Section 2. + +## 4. Our application satisfies the restricted hypothesis + +Both circuits applied in GAUGEFREE-INTERIORS Section 5 are circuits of the normalised kernel +G'_s(u) = exp(-gamma u^2 + 2 gamma s u) (broad: gamma = b0 lambda; narrow: gamma = a rho), whose diagonal is G'_s(0) = 1; +the narrow block itself has N_ii = 1 (Lemma 2.2 at h = 0). The anchored chirps and the constants e^{gamma s^2} are +outside the circuit (diagonal factors, exact anchor ratios). Hence the root kernel has unit diagonal, Lambda >= 1, mu <= 1, +and the inherited parameter values (Section 3 of GAUGEFREE: Lambda <= 3 + sqrt(2 pi/gamma), mu >= p^{-O(1)} for +theta^-1 = O(p^2), the fine-prime range of PR1786) give log(2 + Lambda + 1/mu) <= log 2 + log(2 + Lambda/mu) = O(log p), +i.e. **O(log N log p) guard bits per axis, exactly as stated**. `check_accretive_scale.py` stage E checks symbolically +G'_s(0) = 1 and N_ii = 1, and on exact rational truncated/rounded sections (gamma in {1/2, 2, 20}, s in {0, 1/4, +-0.499}, +n in {2, 4, 8}) that the diagonal is exactly one, e0* H e0 = 1, ||H e0||^2 >= 1 and the Hermitian part is positive definite +(exact LDL^T). + +## 5. Effect on numbers and on the claim + +**None.** The circuit guard enters the ledger only as an additive g * polylog(p) bit term inside the o(p) precision +budget; `outer/check_ledger.py`, `outer/independent_check.py`, `LEDGER.json` and `INDEPENDENT.json` have no Lambda or mu +input, and the binding exponent inequality beta + 2 xi - eps <= 1 - r - zeta is unchanged. `outer/run_outer.py` +regenerates LEDGER.json / INDEPENDENT.json byte-identically after this change; kappa is unchanged. The correction +changes which form of the theorem the package relies on (Corollary 4.1'' / Theorem 4.1'), not any exponent or constant +of the ledger. The rest of the shared PR1673 / PR1758 / PR1786 analytic and fixed-tape review is unaffected by this note +and remains open, as stated in README.md. + +Credit: the counterexample and both repairs are the maintainer's (PR1758 review); Theorem 4.1 and its proof are PR1758's, +resting on PR1673 (Vlado Vojdanovski) SHALLOW-TOEPLITZ.md. + +Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0. diff --git a/research/selective-borrowing-outer-retune/outer/B.json b/research/selective-borrowing-outer-retune/outer/B.json new file mode 100644 index 0000000000..420832abdb --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/B.json @@ -0,0 +1,19 @@ +{ + "b": "16834125885426074623303197403999369605037097224511887913149978507994915321480465896854810510976595214874528056352965500680563625479867406499683286271072676207879008142619299713965094325926283524994199999999999999999999999999999999999999999999999999999999999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "b_above_endpoint": "168341258854260746233031974039993696050370972245118879131499785079949153214804658968548105109765952148745280563529655006805636254798674064996832862710726762078790081426192997139650943259262835249943/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "profile": { + "1": 176252, + "2": 30096, + "3": 30908, + "4": 10152, + "5": 4493, + "6": 18, + "7": 494, + "8": 576, + "32": 11, + "39": 7, + "48": 2493 + }, + "live": 7168, + "note": "b = certified PR2260 complex endpoint (ENDPOINT.json, conservative moment) - 10^-(DIG+80), DIG=200; b_above_endpoint = endpoint + 10^-DIG (rejected)" +} \ No newline at end of file diff --git a/research/selective-borrowing-outer-retune/outer/CLASSICAL-BOUNDARY.md b/research/selective-borrowing-outer-retune/outer/CLASSICAL-BOUNDARY.md new file mode 100644 index 0000000000..b6563eff93 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/CLASSICAL-BOUNDARY.md @@ -0,0 +1,30 @@ +# Classical-matrix boundary (tau = 2.372864 here; PR1699 used 12/5) + +PR1673 (Vlado Vojdanovski) charges the batched boundary algebra W^(tau-1) where tau is a fixed target exponent for W-square complex matrix +products; its stability lemma only requires omega_C < tau (it is stated there for the supplied omega_C <= 9/4 premise, tau = 19/8). +Chafik Boukhalfa (PR1699, `research/t74-signed554-phase-fusion/CLASSICAL-MATRIX-TRANSFER.md`) observed that Francois Le Gall's published +theorem omega < 2.3728639 (ISSAC 2014, arXiv 1401.7714, Table 1 and Section 6.3) already gives omega_C < 12/5, so the matrix branch needs no +unproven premise. This package adopts that specialisation: the boundary exponent is epsilon - xi + (tau-1)/2 = epsilon - xi + 7/10, the boundary +saving is 3/10 + xi - epsilon, and the ledger constant is c = (3 - tau)/2 = 3/10. The checks `matrix_exponent_above_published_bound` +(12/5 > 2.3728639) and `matrix_exponent_below_five_halves` are in the ledger; the exponent is an external published theorem and is not +re-proved here (neither in PR1699 nor in this package). The boundary row is far above the binding rows, so the final kappa is set by the +main transform = fused Gaussian balance with the sparse-union row carrying the layout slack, exactly as in PR1673's matrix corollary (this paragraph describes the PR1699 configuration; in this package the union row is relaxed by the new radix schedule and the boundary row binds, see the end of this file). + +What changes here relative to PR1699 is purely numerical and uses the freedom already in PR1673's proof: every strict slack (r0, zeta, layout slack, +q gap, stop exponent, decimal floor, backoff) is only required to be positive; they are set to 1e-60080 or below, and b is the certified complex +endpoint minus 1e-60080 instead of a 22-digit truncation minus 1e-30. The native moment test of the supplier is re-enclosed on a 2^-199767 grid +(directed-rounding integer fixed point, atanh/exponential term counts derived from the grid in check_ledger.py): moment(b) < 1 and moment(endpoint + 1e-60000) > 1 are both proven by exact rational enclosures. +Resulting kappa exceeds PR1699's exact kappa 0.001677641731343354564251058287... by about 6.7e-25. + + +## Change in this package + +The target exponent is tau = 2.372864 (just above Le Gall's published 2.3728639; PR1699's lemma only needs omega_C < tau), so c = (3 - tau)/2 = 0.313568. With the union row relaxed by the radix-distributed schedule (outer/UNION-SCHEDULE.md, new in this package) the boundary row binds: boundary saving c + xi - eps = kappa + slack, and kappa is set by the balance epsilon = (1 + q + q(c - s - r - zeta))/(1 + q)^2, xi = epsilon (1 + q) - c + s. The checks `matrix_exponent_above_published_bound` (tau - 2.3728639 = 1e-7 > 0) and `matrix_exponent_below_five_halves` are in the ledger. + +## Superseded in this package + +The square exponent tau = 2.372864 above is replaced by the rectangular-batching effective exponent tau_eff = 2.1567081 (k = 5, outer/RECT-BOUNDARY.md). Le Gall's bound is no longer used in the ledger. + +## Superseded again in pkg7 + +The boundary exponent is now tau_eff = 2 + 1e-60080 from structured solves (outer/STRUCTURED-BOUNDARY.md); the fallback is the k = 5 batching of outer/RECT-BOUNDARY.md. diff --git a/research/selective-borrowing-outer-retune/outer/FINE-PADDING-COMPOSITION.md b/research/selective-borrowing-outer-retune/outer/FINE-PADDING-COMPOSITION.md new file mode 100644 index 0000000000..41efb17c1e --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/FINE-PADDING-COMPOSITION.md @@ -0,0 +1,51 @@ +# Fine source-prime padding on this package's stack, and epsilon -> 1 + +**Credit first.** The mechanism of this note is **PR1786 (chafreaky, Chafik Boukhalfa)**: source primes at relative gap Theta(p^-2) from their dyadic targets, which exist by **Tim Trudgian's** effective prime number theorem (arXiv:1401.2689v2, theorem 2), the resulting setup precision `Cp + 64 p^2 ceil(log2 p)`, the theta-dependent conditioning, the polynomial halo p^48 / core p^128 and the dyadic normalizers. Its full text is vendored verbatim in `vendor/pr1786/FINE-PRIME-PADDING.md` and is the proof of every statement about primes, precision, locality and normalization used here. It builds on Vlado Vojdanovski's PR1673 (capped phase-cell fusion), the fractional-shift interiors and per-visit tag accounting of PR1758 (`outer/GAUGEFREE-INTERIORS.md`), the radix-distributed union schedule (`outer/UNION-SCHEDULE.md`), Jeremy Kothe's PR1731 ledger lineage and the PR1699/PR1672 supplier. **New in this package** (only): (a) the observation that epsilon is a free parameter and the resulting exact point epsilon = 1 - 1.0009765625e-60080; (b) the composition analysis below; (c) the exact ledger and independent derivation at that point. + +## 1. What is used from where + +| ingredient | source | role at epsilon -> 1 | +|---|---|---| +| fine primes, theta in (1/(4p^2), 1/(2p^2-1)] | PR1786 sec. 1-2 | chirp range a theta B^2 = O(a B^2/p^2): guard beta + 2 xi - 2 < 1 | +| setup precision, locality, normalizers | PR1786 sec. 3-5 | halo 48 / core 128; online precision stays O(p) | +| fractional-shift interiors, accretive circuit lemma | PR1758 (outer/GAUGEFREE-INTERIORS.md) | removes the linear gauge; **still required**: the old linear guard beta + xi <= 1 is violated at beta ~ 1, xi = 0.908 (ledger control `old_linear_guard`) | +| radix-distributed union schedule, per-visit tag work | outer/UNION-SCHEDULE.md | **still required**: the old g-factor union row 1 - eps - beta - 1/2 + xi is negative (control `old_g_factor_union_row`) and 2 beta < 1 fails (control `old_union_tag_2beta`) | +| boundary | explicit Strassen, tau = 45/16 (7^16 < 2^45), c = 3/32 | **not** the structured boundary of this package series: at xi = 0.908 < eps the boundary row is not binding, so the unproved stability claim of STRUCTURED-BOUNDARY.md is dispensable. Its file is retained as an alternative; it does not enter kappa. | + +## 2. epsilon is a free parameter (answer to "what sets epsilon = 0.9999999 in PR1786") + +Write delta = 1 - epsilon, beta = 1 - 3 delta, xi = epsilon(1+q) - c + s (boundary saving = main + s), c = 3/32. Every constraint of the ledger that involves epsilon or beta: + +* main = epsilon q is the binding saving, kappa = floor(epsilon q) - 1e-60022; +* Gaussian row: 1 - epsilon + q beta >= epsilon q iff **beta >= 1 - delta(1 + 1/q)** (a window of width ~ 594 delta below 1); +* membership: 1 - beta - 2(1-epsilon) > 0 iff **beta < 1 - 2 delta**; header / tag per visit: 1 - beta > 0; group below dimension: epsilon - beta > 0; +* **1 - epsilon - r0 > 0** (`global_gamma_power_slack`, r0 = 1e-60080: the spare power for u <= p^r0) and `source_axis_superpolynomial` 1 - epsilon > 0; +* boundary / union rows: xi = 0.9079 < epsilon (cap_below_cell), xi - 1/2 > 0, guard 1 - (2 xi - 2) - r0 - beta = 0.184 > 0: independent of delta; +* core-face corrections epsilon + beta + 48 - 128 + 2 < 0: independent of delta. + +The window for beta is non-empty for every delta > 0, and beta = 1 - 3 delta lies in it (the Gaussian row exceeds main by delta(1 - 2q)). The only lower bound on delta is **delta > r0 = 1e-60080**, a free strict slack of the proof. So PR1786's 1e-7 is an explicit interior point, not a limit; its kappa is q - 1.68e-10 only because delta = 1e-7. In this package delta = 1.0009765625e-60080 = (1025/1024) 1e-60080 (the smallest ratio that keeps the gamma spare power 1 - epsilon - r0 = r0/1024 > 0 and puts C - epsilon q below 1/F): + + epsilon = 1 - 1.0009765625e-60080, beta = 1 - 3.0029296875e-60080, xi = 0.90792965478815..., + kappa = 0.0016797549561255445213356709939207025769983114385025802954603275201628645... + +which is q - 1.68e-60083 (q = b - 1e-60080 = 0.00167975495612554452133567099392070257699831143850258029546032752016286...). Exact sweep (`outer/independent_check.py`): delta in {1e-7, 1e-12, 1e-30, 1e-60, 1.0009765625e-60080, 1.1e-60080 ... } all satisfy every row; delta = r0 is rejected by `global_gamma_power_slack`; beta below the Gaussian window or at the membership edge is rejected (ledger controls `beta_below_gaussian_window`, `beta_above_membership_window`, `epsilon_equals_one`, `epsilon_not_above_r0_slack`). + +**Maximum epsilon.** Within this construction epsilon < 1 strictly (epsilon = 1 would mean d = p axes: source axes of O(1) size, `epsilon_equals_one` control; also kappa = epsilon q <= q), and 1 - epsilon > r0. The hard cap on any outer of this form is kappa < q (the supplier's complex saving); the remaining distance is q delta = 1.68e-60083. + +## 3. Does composing with the rest of the stack relax anything further? (answer: no, nothing is left to relax) + +* The binding rows at this point are main = epsilon q; boundary = main + s and union = main + s + 13/32, all other rows have positive margins >= 1e-60080. Nothing but delta moves kappa. +* Structured boundary / k5 boundary: boundary row non-binding, no effect on kappa (only on xi, which is free in (1/2, epsilon)). Using c = 1/2 would give xi = 0.5 and the same kappa; it would add the unproved structured-stability claim for nothing. +* The router row a_eff = (q+b)/2 > epsilon q by 5e-65 + q delta; using the true bit endpoint there would not change kappa (main binds). +* Therefore kappa = epsilon q - 1e-60022 with epsilon the only free quantity, and it is already within q delta = 1.68e-60083 of the ceiling q. + +## 4. Obligations (same as PR1786 sec. 8, with sharper asymptotics) + +1. Every "eventually" in vendor/pr1786/FINE-PRIME-PADDING.md is now needed with exponent margin delta = 1.0009765625e-60080 instead of 1e-7: e.g. log x = Theta(p^(1-epsilon)) = p^(1.0009765625e-60080) must dominate log p in the prime-count estimate (true for p beyond an astronomically large cutoff), source axes 2^(p^delta) must exceed every fixed polynomial of p, and the polylog factors of every row are absorbed by p^delta-type slack. This is the same logical status as the 1e-60080 slacks of every row of this package (and of PR1786's 1e-7 version with a smaller cutoff); no practical cutoff is claimed. +2. Trudgian's theorem is a cited mathematical input (vendor/pr1786); the prime and precision lemmas are PR1786's and need their review (PR1786 is a draft). +3. The composition with PR1758's accretive circuit lemma uses theta^-1 = O(p^2) in the symbol bound (PR1786 sec. 3); the Gauge-free note's guard is superseded by `beta + 2 xi - 2 < 1` at fine theta. +4. Fresh native admission of the unchanged supplier: `verify_release.py`. + +## 5. Executable support + +`outer/check_ledger.py` (exact, 62 strict checks (including improvement over the exact kappas of PR1828, PR1839, PR1875, PR1876 and PR1897), 17 rejecting controls, improvement over PR1786's exact kappa and PR1758), `outer/independent_check.py` (independent balance derivation, the epsilon-free sweep), `outer/check_fine_padding.py` (PR1786's finite prime witnesses, underflow control, power relations), `outer/check_gaugefree.py` (PR1758 identities). diff --git a/research/selective-borrowing-outer-retune/outer/GAUGEFREE-INTERIORS.md b/research/selective-borrowing-outer-retune/outer/GAUGEFREE-INTERIORS.md new file mode 100644 index 0000000000..91b0200829 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/GAUGEFREE-INTERIORS.md @@ -0,0 +1,121 @@ +> **Package note (scale qualification, #2265).** As a uniform bound in Lambda and mu, Theorem 4.1 below is false (N = 2, H = tI; maintainer review https://github.com/CrocSwap/integer-mult-bounds/pull/1758#issuecomment-6203527654). The text below is kept verbatim; this package relies on the corrected form in `outer/ACCRETIVE-SCALE.md`: in general log(2 + Lambda + 1/mu) replaces log(2 + Lambda/mu) (as in SHALLOW-TOEPLITZ sec. 9), and the ratio form holds for unit-diagonal kernels (Lambda >= 1, mu <= 1, so Lambda + 1/mu <= 2 Lambda/mu), which covers every kernel G'_s applied here, including truncated/rounded kernels and the descendant/shear and boundary-pivot bounds. No ledger number changes. + +> **Package note (fine primes).** In this package the source primes are chosen at relative gap Theta(p^-2) (PR1786, `vendor/pr1786`); theta >= 1/(4p^2) replaces theta >= 1/(4d) in every bound below, the chirp range becomes O(a B^2/p^2) and the guard of Section 5 becomes beta + 2 xi - 2 < 1 (see `outer/FINE-PADDING-COMPOSITION.md`). The accretive lemma and the shift identities hold for every theta > 0; the symbol bound mu >= (1 - e^{-gamma theta/2}) e^{-pi^2/(3 gamma)} is polynomial for theta^-1 = O(p^2). + +# Gauge-free interiors: fractional-shift Toeplitz kernels, accretive shallow circuits, and the chirp-only guard + +**Status.** New in this package series (not in PR1673 by Vlado Vojdanovski, whose capped phase-cell fusion, SHALLOW-TOEPLITZ.md, CAPPED-GAUGE-AUDIT.md and LOG-A-PHASE-SCHUR-AUDIT.md this note modifies and relies on). Conditional on every hypothesis of PR1673 except the three statements replaced below. Independent review required before release. Executable support: `outer/check_gaugefree.py` (exact identities, 50-digit symbol checks, section/Schur bounds, circuit replay) and `outer/check_ledger.py` / `outer/independent_check.py` (the ledger). + +## 0. Result and what changes + +kappa (this package = structured boundary + this note) = **0.0016797455088022990515465336851999736723...** (exact, 58 strict checks positive, all rejecting controls fire; previous package (structured boundary, linear guard) 0.0016783441717365..., gain +1.4013e-6; rectangular k=5 package 0.0016781238306335..., gain +1.6217e-6). Parameters: eps = 0.9999943758, xi = 0.5016741213, beta = 0.9966461332. + +Three statements of #1673 are replaced (everything else is unchanged): + +| # | #1673 / earlier packages | this note | +|---|---|---| +| R1 | local gauge of a capped interior: diagonals of range O(aB) (narrow), O(b0 B) (broad) | the linear part of the gauge is exactly a fractional centre shift of the Toeplitz kernel; only the quadratic chirp remains, range O(a theta B^2) = O(a B^2/d). (Section 2) | +| R2 | SHALLOW-TOEPLITZ sec. 9 needs mu I <= H <= M I (SPD) or ||H-I||<1/4 | accretive: (H+H^*)/2 >= mu I and ||H|| <= Lambda. The shifted Gaussian symbol is accretive with explicit mu. (Sections 3, 4) | +| R3 | ledger guard beta + xi <= 1-r-zeta; union tag check 1-2 beta > 0 | guard beta + 2 xi - eps <= 1-r-zeta; tag check 1-beta > 0 (Sections 5, 6) | + +## 1. Setting + +Phase cell: beta_i = theta i - c in [-1/2,1/2), rho = 1+theta, lambda = 1/rho, 1/(4d) <= theta <= 1/(2d-1). An interior is a contiguous block of selected target coordinates (unit q-gaps), length n <= B (dyadic), anchor m at its centre; x = i-m, y = k-m, t = i-m for the narrow operator. B = p^xi, d = p^eps, xi < eps. + +* Broad forward block (FORWARD-SMALL-A-AUDIT, CAPPED-GAUGE-AUDIT sec. 3): K_{ki} = alpha^-1 exp(-b0 (x - lambda y + delta)^2), delta = (1-lambda) m - c, |delta| <= 1/2, b0 = pi/alpha^2. +* Narrow inverse block (LOG-A-PHASE-SCHUR-AUDIT sec. 1): N_{i,i+h} = exp(-a[(rho h + beta_i)^2 - beta_{i+h}^2]), a = pi alpha^2, beta_i = beta0 + theta t_i. + +## 2. Exact gauge <-> shift identities + +**Lemma 2.1 (broad).** For every real s, +(x - lambda y + delta)^2 - lambda (x - y - s)^2 = (1-lambda) x^2 - lambda(1-lambda) y^2 + delta^2 - lambda s^2 + 2(delta + lambda s) x - 2 lambda (delta + s) y. +Hence K = Drow G_s Dcol exactly, with the Toeplitz kernel G_s(y,x) = exp(-b0 lambda (x-y-s)^2) and diagonals separable in (y,x) (proved by expansion; sympy and 600 exact rational mixed-difference checks in `check_gaugefree.py` stage A/B). +Choose **s = -delta (1-theta)**. Then the x-linear coefficient is 2 b0 (delta + lambda s) = 2 b0 delta (1 - lambda(1-theta)) = 2 b0 delta * 2 theta/(1+theta), |.| <= 2 b0 theta, and the y-linear coefficient is -2 b0 lambda (delta + s) = -2 b0 lambda theta delta, |.| <= b0 theta. Over a block with |x|,|y| <= B/2 the linear leftover has logarithmic range <= 2 b0 theta B (total), and the quadratic parts b0 (1-lambda) x^2, b0 lambda (1-lambda) y^2 have range <= b0 theta B^2/4 each. Also |s| <= (1-theta)/2, i.e. 1 - 2|s| >= theta. +(With s = -delta/lambda the linear terms vanish in x and are 2 b0 delta (1-lambda) in y, but |s| can exceed 1/2 by theta/2, which changes the winding number of the symbol; so the shift is shrunk by (1-theta). The price is the O(b0 theta B) linear leftover, dominated by the chirp.) + +**Lemma 2.2 (narrow).** With beta_{i+h} = beta_i + theta h, rho = 1+theta: +(rho h + beta_i)^2 - beta_{i+h}^2 = (1+2theta) h^2 + 2 beta_i h. Writing beta_i = beta0 + theta t_i, j = i+h, t_j = t_i + h, and 2 t_i h = t_j^2 - t_i^2 - h^2: +log N_{ij} = -a rho h^2 - 2 a beta0 h + a theta (t_i^2 - t_j^2) (exact; sympy check, stage A). +So N = D' k D'^{-1}-type: N_{ij} = e^{a theta t_i^2} k(j-i) e^{-a theta t_j^2} with the Toeplitz kernel k(h) = exp(-a rho h^2 - 2 a beta0 h) = exp(a beta0^2/rho) exp(-gamma (h-s)^2), **gamma = a rho, s = -beta0/rho**. Here |s| <= 1/(2 rho) so **1 - 2|s| >= 1 - 1/rho = theta/rho >= theta/2**: no shrinking is needed, and the gauge is the chirp D' = exp(a theta t^2) of logarithmic range a theta B^2/4 (centred). The old weights are the same objects: log w_i = const + a theta (i - c/theta)^2 (the old linear slope 2 a beta0 is exactly the term moved into k). + +**Normalisation.** G'_s(u) := e^{gamma s^2} e^{-gamma (u-s)^2} = exp(-gamma u^2 + 2 gamma s u) has G'_s(0) = 1 (unit diagonal, as the audited N). The scalars e^{gamma s^2} are exact constants (stored in the anchor ratios). + +**Rounding.** The circuit applies the ideal truncated kernel (band W) rounded to Cp bits; the audited global rounded matrix differs from it by <= poly(p) 2^{-Cp} times the diagonal range e^{O(g (a theta B^2 + polylog))}, which is covered by the same precision guard (Section 5). **Restated comparison (audited rounded matrix vs ideal shifted kernel).** The audits (BILATERAL-LDU, FORWARD-SMALL-A, CAPPED-GAUGE-AUDIT sec. 2-3) bound the unshifted globally rounded matrix H^. Let K be the ideal block (exact Gaussian entries) and K~ = Drow' G^_s Dcol' the matrix the circuit applies (truncated at W, G^_s rounded to Cp bits, anchored diagonals). Entrywise |H^ - K| <= 4 * 2^{-P} e^{a/(4 theta)} relative to the weights (audit sec. 2), and |K~ - K| <= 2^{-Cp} W-banded; the diagonal conversions contribute the factor e^{O(a theta B^2)} per axis (chirp range of Section 5; the linear leftover is smaller). Hence ||K~ - H^|| <= poly(p) 2^{-Cp} e^{O(g a theta B^2 + g polylog p)} (g axes tensored), and the exponent is o(p) bits by the precision guard beta + 2 xi - eps < 1 of Section 5. For C > 1 the perturbation is <= 2^{-(C-o(1))p} <= 2^{-p}/poly, and since ||K^-1||, ||H^-1|| <= p^{O(1)} (Section 3; audits) every audited uniform bound (row gap, inverse norms, Neumann tail J = Cp^2, halo radius A) holds for K~ with polynomially perturbed constants. No structural change. + +## 3. The shifted Gaussian symbol is accretive + +Let q = e^{-gamma}, 0 <= s' = |s| < 1/2, phi(w) = sum_u e^{-gamma(u-s)^2} e^{-iuw}. + +**Lemma 3.1 (product form).** Jacobi triple product with Z = q^{-2s} e^{-iw}: +phi(w) = e^{-gamma s^2} prod_{m>=1}(1-q^{2m}) (1 + a_m e^{-iw})(1 + b_m e^{iw}), a_m = q^{2m-1-2s}, b_m = q^{2m-1+2s}. +For s < 0 the symbol is the complex conjugate (swap a, b). All a_m, b_m in (0,1) iff |s|<1/2; a_1 = q^{1-2|s|} is the unique factor near 1. At |s| = 1/2: a_1 = 1 and phi(pi) = 0. (Verified at 50 digits, relative error 2.5e-42 on 1025 points.) + +**Lemma 3.2 (phase).** For w in [0,pi] and s >= 0, f(x) := arg(1 + x e^{iw}) = atan(x sin w/(1 + x cos w)) is increasing in x in [0,1) (f'(x) = sin w/|1+x e^{iw}|^2 > 0). arg phi = sum_m [f(b_m) - f(a_m)]. Since 2m+1-2s >= 2m-1+2s iff s <= 1/2, a_{m+1} <= b_m, so f(b_m) >= f(a_{m+1}) and +0 <= -arg phi = sum_m [f(a_m)-f(b_m)] <= sum_m [f(a_m) - f(a_{m+1})] = f(a_1) < w/2 <= pi/2. +For w in [-pi,0] use phi(-w) = conj(phi(w)). + +**Lemma 3.3 (real part).** cos(f(a_1)) = (1 + a_1 cos w)/|1 + a_1 e^{iw}|, so +Re phi(w) >= |phi(w)| cos f(a_1) >= e^{-gamma s^2} prod_{m>=1}(1-q^{2m}) * (1 - a_1) prod_{m>=2}(1-a_m) prod_{m>=1}(1 - b_m) =: m_s(gamma). +(The bound is attained at w = pi up to the other factors: stage C finds min Re phi / m_s = 1.0.) Normalising by e^{gamma s^2}: + +mu(gamma, s) := m_s e^{gamma s^2} >= (1 - e^{-gamma(1-2|s|)}) exp(-pi^2/(3 gamma)). + +Proof of the closed form: -ln prod_k (1 - e^{-k gamma x}) = sum_h (1/h)/(e^{h x gamma}-1) <= pi^2/(6 x gamma); the three infinite products are bounded by prod_{k>=1}(1-q^{2k}) >= e^{-pi^2/(12 gamma)} (twice, the second for m >= 2 since a_m <= q^{2m-2}) and prod_{k>=1}(1-q^{2k-1}) >= prod(1-q^k) >= e^{-pi^2/(6 gamma)}; total e^{-pi^2/(3 gamma)}. +Norm: Lambda := ||G'_s||_2 <= ||symbol||_{l1} = sum_u e^{-gamma(u^2 - 2su)} <= 3 + sqrt(2 pi/gamma) (u^2 - 2su >= u^2 - |u| >= 0 and >= u^2/2 for |u| >= 2). + +Parameter values. Broad: gamma = b0 lambda = pi/(alpha^2 rho) = Theta(1/log p), 1-2|s| >= theta >= 1/(4d): mu >= (gamma theta/2) e^{-pi alpha^2 rho /3} = p^{-O(1)}. Narrow: gamma = a rho >= 4 log(16 p), 1 - 2|s| >= theta/2: mu >= (1 - e^{-a theta/2}) e^{-pi^2/(3 gamma)} >= a theta/3 = p^{-O(1)} (this is the audited row gap). + +## 4. The accretive Toeplitz circuit (replaces SHALLOW-TOEPLITZ sec. 9) + +**Theorem 4.1.** Let N be a power of two and H an N x N complex Toeplitz matrix with ||H||_2 <= Lambda and (H + H^*)/2 >= mu I, mu > 0. There is an N-wire circuit of disjoint invertible 1x1/2x2 gates applying H, with O(log^4 N) rounds, O(N log^4 N) gates, no ancilla, and +log(max prefix / suffix / inverse-prefix operator norm) = O(log N * log(2 + Lambda/mu) + log N). +Rounding every coefficient to p + O(log N * log(2+Lambda/mu)) bits gives operator error 2^-p. If Lambda/mu = p^{O(1)} the guard is O(log N log p) bits per axis. + +*Proof.* Identical to SHALLOW-TOEPLITZ sec. 4, 5, 6, 9 except for the single place where the hypothesis is used, the uniform control of every recursive matrix, which we prove now. + +(a) *Descendants are Schur complements of sections.* Let T_b be the leading b x b section of H. The recursion of sec. 4 acts on index intervals J = [a,b); the matrix at J is S(a,b) = T_b / T_a (Schur complement of the leading a x a block of T_b, as an (b-a) x (b-a) matrix). Indeed the root is S(0,N); the children of S(a,b) with midpoint m are S(a,m) = T_m/T_a (leading block of S(a,b)) and S(a,b)/S(a,m) = T_b/T_m by the quotient property (T_b/T_a)/(T_m/T_a) = T_b/T_m. All of these are contiguous leading-type sections of the same Toeplitz H. + +(b) *Accretive matrices.* If Re >= mu ||x||^2 then ||Tx|| >= mu||x||, so ||T^-1|| <= mu^-1; every principal submatrix has the same property. Hence ||T_b^-1|| <= 1/mu for all b and ||S(a,b)^-1|| = ||(T_b^-1)[a,b)|| <= 1/mu. Also ||S(a,b)|| <= ||T_b[a,b)|| + ||T[a,b); [0,a)|| ||T_a^-1|| ||T[0,a); [a,b)|| <= Lambda + Lambda^2/mu =: Lambda'. The shear matrices K = A^-1 B, L = C A^-1 have norm <= Lambda/mu. (No Hermitian structure was used.) + +(c) *Displacement rank, generators, compiler.* Sec. 4's rank identities (Delta = ZM - MZ, rank Delta(H) <= 2, ranks of A,S <= 2 + depth) and the sec. 5 bounded-generator compiler use only that each node's A, S, K, L are bounded matrices with the displayed ranks; the Stein generators g_j (orthonormal), h_j (norm <= ||E_T|| <= 2 Lambda'') have polynomial norm, and gamma_P = 64 N Lambda'' suffices. The recursion D_full(N) <= D_full(N/2) + O(log^3 N) is unchanged. + +(d) *Prefix norms.* With each completed node factor of norm and inverse norm <= poly(N) (Lambda'/mu)^{O(1)}, B(N) <= max(C B(N/2), N^C (Lambda'/mu)^C), so log B(N) = O(log N log(2+Lambda'/mu)) as in sec. 9 with kappa replaced by Lambda'/mu. QED. + +**Corollary 4.2 (perturbation, truncation, rounding).** If the symbol is perturbed in l1 by eps' <= mu/2 (band truncation at W, rounding to Cp bits), the Hermitian part stays >= mu/2 and Theorem 4.1 applies to the perturbed matrix. For the broad kernel the truncation tail is <= 2^{-p}, and mu >= p^{-O(1)}. + +**Corollary 4.3 (boundary pivots for the structured boundary).** The boundary LU pivot P_j of STRUCTURED-BOUNDARY (Schur complement of T[J] onto segment j after eliminating all indices left of it and the interval I_j to its right, J = [0, end of I_j)) satisfies ||P_j^-1|| <= 1/mu, ||P_j|| <= Lambda + Lambda^2/mu, because P_j^-1 is the principal block of (T[J])^-1 and T[J∖seg_j] is a principal submatrix of T[J], both accretive. The couplings are band entries of G_s. The statement holds per constant-offset stretch: each stretch of a cell (maximal run with constant phase offset c, i.e. one Toeplitz matrix T, one shift s) is handled with its own anchor at the segment centre, so that the chirp D' = exp(a theta t^2) (broad: b0 theta t^2) has range a theta B'^2/4 for boundary segments of physical length B' <= B + 2W (and the linear leftover is <= 2 b0 theta B'); a pivot chain crossing a cell cut restarts, since cell cuts are boundary coordinates. Rank-2 Stein/Sylvester displacement for P_j, P_j^-1 holds for every Toeplitz T (Kailath), shifted or not; screened in `check_gaugefree.py` D/E and in the earlier IdeaOuter screen (displacement rank 2 along the whole chain at s = 0.5). Thus the structured-boundary pivot bounds (which used BILATERAL-LDU for the unshifted gauge) transfer without change of statement. + +## 5. Assembly and the guard + +For every capped interior (dyadic length n <= B, with its own anchor and shift s): + broad: Drow' (circuit of G'_s, Theorem 4.1 with mu = mu(b0 lambda, s)) Dcol'; + narrow: D' (circuit of G'_s, gamma = a rho) D'^-1 (inverse circuit = reversed gate sequence). +The diagonal factors Dcol', Drow', D' are anchored ratios with logarithmic range + narrow: a theta B^2/4; broad: <= b0 theta B^2/4 (x^2 and y^2 terms) + 4 b0 theta B (linear leftover) . +Per axis the range is O(a B^2/d); direct sums over blocks run in parallel as before. For g simultaneous axes the precision requirement of CAPPED-GAUGE-AUDIT sec. 4 becomes + P >= p + O(a d) + O(g a theta B^2) + O(g log N log p) + ... , i.e. g B^2/d * polylog <= p^{1-r-zeta}: + **beta + 2 xi - eps <= 1 - r - zeta**. +The offline accuracy term a d = o(p) is unchanged. The leftover linear term (exponent beta + xi - eps) is dominated by the chirp (exponent beta + 2 xi - eps) since xi > 0. The circuit norms add g * O(log N log p) bits (Theorem 4.1), polylog per axis, as before. +(Why this does not collide with locality: the audited locality/halo constructions use the global rounded matrix only through uniform norms and the row gap, which are properties of the same Toeplitz-similar matrices; the shift is an exact similarity-to-Toeplitz representation of the same blocks, Lemmas 2.1-2.2.) + +## 6. Ledger changes (exact) + +1. `group_gauge_power_slack`: 1 - xi - r - beta -> **1 - (2 xi - eps) - r - beta**; `actual_logarithmic_group_guard`: 1 - xi - beta -> **1 - (2 xi - eps) - beta**. Add `shift_leftover_linear_dominated_by_chirp`: xi > 0, `shift_gap_theta_polynomial`: eps > 0. +2. Parameters: eps = (1 + q beta)/(1 + q) (main = Gaussian), xi = eps(1+q) - c + slack (boundary = main + slack), beta = 1 - 2 xi + eps - r - zeta. Closed form beta = (-1 + 2c - 2 slack + 1/(1+q) - r - zeta)/(1 + 2q - q/(1+q)), c = (3 - tau_eff)/2. +3. `union_tag_work_fits_record` (1 - 2 beta) -> **`union_tag_work_per_visit_fits_record`: 1 - beta** (Section 7). `union_header_fits_record` (1-beta) and `membership_arithmetic_fits_record` (1 - beta - 2(1-eps)) stay: pkg8 values 3.354e-3 and 3.343e-3. +4. Controls added: `old_linear_gauge_guard` (1 - xi - beta < 0), `old_union_tag_2beta` (1 - 2beta < 0) must fire as rejecting controls (they do). +5. Numbers (pkg8-era package, superseded by outer/FINE-PADDING-COMPOSITION.md for the current epsilon = 1 - 2e-100 point): eps = 0.99999437577296, xi = 0.50167412128176, beta = 0.99664613320944; margins above kappa: main = Gaussian = bit router = boundary = union ~ 1e-64; global router 9.4e-9; cap xi - 1/2 = 1.674e-3; group_below_dimension 3.35e-3; epsilon_below_one 5.6e-6; improvement over pkg7 1.4013e-6. +`outer/check_ledger.py` carries exactly these edits relative to the previous (structured-boundary, linear-guard) package; `outer/independent_check.py` re-derives (E,X,B) from the three linear equations by exact elimination. + +## 7. Union tag work: g polylog <= p suffices + +The previous version of UNION-SCHEDULE.md (steps 1 tag and section 3 header paragraph) argued "tag work per record <= O(g^2) in total, o(p) because 2 beta < 1". The cost accounting of Section 2.3-2.4 of that note is per visit: round r touches only records of S_r; each visit costs O(p log g + p log a + 1) head travel (radix passes, transposition) plus the tag extraction O(g) bit operations (scan of the g-bit flag header for the r-th set bit) plus O(g log L) header travel (the header is carried with the record). Sum over rounds = sum_r |S_r| = sum_x |F(x)| <= g mu V (Section 2.4). Total = O(g mu V (p polylog p + g log L)) = O(g mu V p polylog p) as soon as g log L = o(p polylog p), i.e. beta < 1 (and this is already the header check `union_header_fits_record`). The O(g^2) per-record total is a pessimistic restatement: each of the |F| <= g visits of a record is paid for by its own Theta(p) move. UNION-SCHEDULE.md now carries this wording. A reviewer should confirm no other place charges g^2 per record. + +## 8. Obligations / risks (review checklist) + +1. Theorem 4.1 relies on SHALLOW-TOEPLITZ sec. 4-6 (not re-derived); only the descendant bound (Section 4(a)-(b)) is new. The finite replay (`check_gaugefree.py` stage E: the vendored compiler on shifted Gaussian tiles n = 8, 16, 32, gamma in {0.5, 2, 20}, s up to 0.499: max error 1.8e-12, displacement rank <= 3, peak prefix norm <= 91 for n = 8, 16, max generator entry <= 11.4) supports it. The peak prefix norm is tracked only for n <= 16; for n = 32 it is not computed (reported as null). +2. The tile is a contiguous section of the *banded, truncated, rounded* shifted kernel; Corollary 4.2 covers it. The audited global rounded matrix remains the reference for the locality/Neumann/halo arguments; its comparison with the ideal kernel is stated in Section 2 (error poly(p) 2^{-Cp} e^{O(g a theta B^2)} = 2^{-(C-o(1))p}, C > 1). +3. Narrow N formula taken from LOG-A-PHASE-SCHUR-AUDIT (`n_{i,h}`), verified symbolically against the quoted weights of CAPPED-GAUGE-AUDIT sec. 2 (stage A). Broad block formula from CAPPED-GAUGE-AUDIT sec. 3. +4. (pkg8-era, superseded by the fine-padding point beta = 1 - 6e-100) The parameter point sat at beta = 0.9966, 1 - beta = 3.35e-3: header (g log L bits) and membership (g (log s)^2) rows hold only "eventually", with exponent margins 3.3e-3; the claim is an asymptotic statement with the same status as the other margins (1e-64 slacks). +5. Not changed: supplier, bit/complex rows, union schedule beyond Section 7, cap xi > 1/2 (still binding at 1.67e-3). +6. STRUCTURED-BOUNDARY.md now states the chirp range O(aB^2/d) for consecutive boundary segments (physical length <= B + 2W) and cites Corollary 4.3 for the pivot bounds. diff --git a/research/selective-borrowing-outer-retune/outer/INDEPENDENT.json b/research/selective-borrowing-outer-retune/outer/INDEPENDENT.json new file mode 100644 index 0000000000..c450b6d88d --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/INDEPENDENT.json @@ -0,0 +1,43 @@ +{ + "status": "PASS_INDEPENDENT_BALANCE_DERIVATION", + "source_sha256": "c60d04a8d971d0e984aeeeda87ae613d712537b2623e57a2f315983245e8a280", + "b": 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"5000000000000000000000004999999999999999999999999999999999/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "epsilon_free_sweep_delta_gap_to_q": [ + [ + "1/10000000", + "1.6834125885426075e-10" + ], + [ + "1/1000000000000", + "1.6834125885426075e-15" + ], + [ + "1/1000000000000000000000000000000", + "1.6834125885426075e-33" + ], + [ + "1/1000000000000000000000000000000000000000000000000000000000000", + "1.6834125885426075e-63" + ], + [ + "41/409600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "1.685056546148606e-283" + ], + [ + "11/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "1.851753847396868e-283" + ] + ] + } + }, + "scope": "Independent balance derivation and margin comparisons for this package's supplier endpoint; the supplier native admission is bound by verify_release.py; inherited all-size assumptions remain." +} diff --git a/research/selective-borrowing-outer-retune/outer/LEDGER.json b/research/selective-borrowing-outer-retune/outer/LEDGER.json new file mode 100644 index 0000000000..6f9a081734 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/LEDGER.json @@ -0,0 +1,534 @@ +{ + "status": "PASS_FINE_PRIME_PADDING_EXACT_ARITHMETIC", + "scope": "Conditional fine-prime padding composition. Fresh native admission is required by verify.py. The cited prime theorem, written gauge-free theorem and inherited all-size interfaces are not proved by this arithmetic. No independent human review is claimed.", + "geometry_changes": { + "theta_power": { + "rational": "2", + "decimal": "2" + }, + "halo_power": 48, + "core_power": 128, + "matrix_exponent": { + "rational": "45/16", + "decimal": "2.8125" + } + }, + "complex_lower_short": { + "rational": "16834125885426074623303197403999369605037097224511887913149978507994915321480465896854810510976595214874528056352965500680563625479867406499683286271072676207879008142619299713965094325926283524994199999999999999999999999999999999999999999999999999999999999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "decimal": 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"6252436828226462330319740399936960503709722451188791314997850799491532148046589685481051097659521487452805635296550068056362547986740649968328627107267620787900814261929971396509432592628352499419999999999999999999998999999999999999999999999/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "decimal": "0.000006252436828226462330319740399936960503709722451188791314997850799491532148046589685481051097659521487452805635296550068056362547986740649968328627107267620787900814261929971396509432592628352499419999999999999999999998999999999999999999999999" + }, + "source_sha256": { + "SPEC.json": "c60d04a8d971d0e984aeeeda87ae613d712537b2623e57a2f315983245e8a280", + "ENDPOINT.json": "aefecb3ba73c417e6942fb440fb9f964e74a635dd4d73074ecc54e31768df5be" + } +} diff --git a/research/selective-borrowing-outer-retune/outer/PROOF-GRID-MAX.md b/research/selective-borrowing-outer-retune/outer/PROOF-GRID-MAX.md new file mode 100644 index 0000000000..ac245bc564 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/PROOF-GRID-MAX.md @@ -0,0 +1,15 @@ +# Maximum on the declared fixed kappa grid + +Credit: this argument and its parameter point are PR2180 / PR2161 (dataisfire, with OpenAI Codex assistance) on this series' PR2044; the ledger formulas are PR1786 (chafreaky / Chafik Boukhalfa), PR1673 (Vlado Vojdanovski), PR1758 and the Kothe lineage. Re-derived here at this package's own endpoint. + +Let C be the certified complex supplier endpoint, F = 10^60046 the floor denominator and B = 10^-60022 the positive backoff of `outer/check_ledger.py`. Every point of the free-gap family uses 0 < epsilon < 1 and 0 < q < b < C and reports K = floor(F epsilon q)/F - B. Since C F is an integer (asserted exactly) and epsilon q < C, integer ordering gives + + K <= C - 1/F - B. + +This bounds the declared grid/backoff family, not arbitrary suppliers, other backoffs or finer grids. + +Choose r0 = 10^-60080, zeta = r0/100, layout slack = gap = r0, stop = r0/10^6, b = C - r0, q = C - 2 r0, delta = (1025/1024) r0, epsilon = 1 - delta, beta = 1 - 3 delta. Then 0 < C - epsilon q < 1/F, checked exactly, so the reported K equals the bound C - 1/F - B (asserted in `outer/check_ledger.py`, `outer/independent_check.py`). + +All 62 strict inequalities of the earlier ledger and all 17 rejecting controls are retained. The main-binding family condition is now an explicit 63rd strict row, `gaussian_above_main`: Gaussian saving minus main saving = delta (1 - 2q) > 0. It was previously implicit in the checked beta window; it matters for rejection controls because the kappa floor/backoff (1e-60046, 1e-60022) is coarser than the free gaps (1e-60080), so a beta outside the main-binding window could still exceed the rounded kappa. The gamma spare power is r0/1024 > 0. All eventual cutoffs remain asymptotic; no practical cutoff is claimed. + +The supplier, native words, owners, invoice, endpoint and fine engines are unchanged (this package's own verify.py re-admits them in full). The ledger re-encloses the native moment sum at the new b = C - r0 (moment(b) < 1) and at the test endpoint C + 1e-60000 (moment > 1). diff --git a/research/selective-borrowing-outer-retune/outer/RECT-BOUNDARY.md b/research/selective-borrowing-outer-retune/outer/RECT-BOUNDARY.md new file mode 100644 index 0000000000..c49777f8be --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/RECT-BOUNDARY.md @@ -0,0 +1,72 @@ +> **Superseded in this package (pkg7) by outer/STRUCTURED-BOUNDARY.md; kept as the documented fallback (pkg6: k = 5, tau_eff = 2.1567081) if the structured-solve argument is not accepted.** + +# Rectangular boundary batching, k = 5 (new in this package) + +**Status.** This construction is a new contribution of this package. The capped phase-cell fusion outer, its boundary algebra and the batching of right-hand sides are Vlado Vojdanovski's +PR1673 (PHASE-LAYOUT-AUDIT.md, "Batched right-hand sides"); the specialisation of the matrix boundary to an established published exponent is Chafik Boukhalfa's PR1699; the +radix-distributed sparse-union schedule is new in pkg4 (outer/UNION-SCHEDULE.md). The change here: the boundary solves are batched with **W^k right-hand-side columns per factor page with k = 5** +(pkg5 used k = 2), so that rectangular matrix multiplication is used. The published input is the bound **omega(1,1,5) = omega(1,5,1) = omega(5) < 6.156708** of Francois Le Gall, +"Faster Rectangular Matrix Multiplication by Combination Loss Analysis", SODA 2024, arXiv 2307.06535 (v2, revised 27 Dec 2023), Section 1.2, Table 3 ("Our new upper bounds on omega(kappa)"), kappa = 5.00 -> 6.156708 +(same table: kappa = 3.00 -> 4.199095, kappa = 2.00 -> 3.250563, kappa = 1.00 -> 2.372537). Convention (Section 1.1): omega(kappa) is the smallest exponent such that the product of an n x ceil(n^kappa) matrix by a +ceil(n^kappa) x n matrix is computable in O(n^(omega(kappa)+eps)), i.e. the exponent of the tensor ; footnote 1 there (citing [8, 9]) states that the arithmetic complexity of n x n times n x m, n x m times m x n and m x n times n x n are the same: +the matrix multiplication tensors , , are cyclic rotations of each other, so and (a W x W factor page against W x W^5 right-hand sides) have the same rank and omega(1,5,1) = omega(1,1,5). +The theorem is used as a cited input and is not re-proved here. All inherited all-size hypotheses of PR1673 remain. (The k = 2 value of ADVWXXZ 2024, omega(1,1,2) < 3.250035, is the pkg5 fallback; for k = 5 only Le Gall's table gives a bound.) + +## 1. Batch width and the per-column cost + +In the audit, all inner coordinates and following tensor axes are independent right-hand sides; their number R is superpolynomial. The audit batches them in a dyadic width between W and 2W. Here a batch has width between W^5 and n^5 W^5, +where n is the fixed base of the rectangular format of section 3 (a constant; the width is a power n^(5j) matching the padded row dimension n^j). +Batching more columns is allowed for the same reason as before: the number of right-hand sides in an axis substream is superpolynomial, so padding the last incomplete batch costs a polynomial amount per axis substream +(at most n^5 W^5 lines of |E_f| records; at most g axes and g rounds per window), negligible against the superpolynomial volume. The batch matrix of |E_f| x n^5 W^5 consecutive lines (each line |E_f| long) is +rearranged so that W-row slices times all batch columns are contiguous by another small-field transpose with O(log(n^5 W^5)) = O(log p) riffle passes, so moving it costs O(log p) times the boundary volume, exactly as in the audit. +**Working set and record count per batch:** one batch is the W x n^5W^5 matrix of a block (and its W x n^5W^5 result) of p-bit records: n^5 W^6 = poly(p) records (about p^3 polylog p), so P-bit entries fit in polynomially many tape cells; +the whole batch matrix |E_f| x n^5 W^5 is |E_f| n^5 W^5 records, polynomial in p. **Superpolynomial supply per factor page (premise used explicitly):** every prefix block of an axis substream contains +L^(g-f) times the spectators lines, which is superpolynomial in p (the spectator payload and the L^(g-f) suffixes), hence >> n^5 W^5 lines: there are enough independent right-hand sides per factor page. For the last axis f = g the suffix is empty and the lines of a block are only the spectator lines: the d - g spectator axes (d = p^epsilon axes in total, g = p^beta in the group) each have length L = p^8, so there are L^(d-g) = p^(8(d-g)) of them, superpolynomial because d - g = p^epsilon - p^beta grows; W^5 = p^2.5 polylog p is far below it. The factor pages depend only on the axis f, the block index along the line and the axis-f coarse label, not on the other coordinates, so lines of every prefix block of the tag-f substream share the same pages. Each W x W factor page is scanned once per batch (page cost W^2 per block-batch versus product W^sigma). + +Cost bookkeeping per axis: a band of boundary rows has V W / B right-hand-side entries per axis. A batch handles W * W^5 = W^6 entries with one product of shape costing W^(omega(1,1,5)+eps). +So per axis the boundary arithmetic is (V W / B) / W^6 * W^(omega(1,1,5)) = (V/B) W^(omega(1,1,5) - 5), i.e. the old form (V/B) W^(tau - 1) with +**tau_eff = omega(1,1,5) - 5 + 1 = omega(1,1,5) - 4** (in general tau_eff = omega(1,1,k) - k + 1; k = 1 is the square case tau = omega). With the published bound, tau_eff < 2.156708. +The boundary row exponent is epsilon - xi + (tau_eff - 1)/2 and its saving is c + xi - epsilon with c = (3 - tau_eff)/2. + +## 2. Blocked triangular solve against W^5 columns, geometric ratio + +The no-pivot band solve of the audit is the block recursion X_j = L_jj^-1 (Y_j - L_{j,j-1} X_{j-1}) on W x W blocks along an axis line, with R = batch width columns: Y_j, X_j are W x R. Each step is two products of a W x W factor +block with a W x R matrix: shape with R = n^(5j), W = n^j padded. The precomputed diagonal inverses L_jj^-1 are part of the factor page. +The product is computed by recursion on the fixed format (n x n by n x n^5 matrices, rank R_f): T(W) <= R_f T(W/n) + A(W), where the additions (linear combinations of blocks and accumulations) at the top level touch +O(W^6) entries (the output W x W^5 and the inputs are of size at most W^6): A(W) = O(W^6). The output size of the format is n * n^5 = n^6 entries, so necessarily R_f >= n^6; the format exponent sigma > 6 satisfies R_f < n^sigma. The level-i additions cost +R_f^i (W/n^i)^6 = W^6 (R_f/n^6)^i, a geometric series with ratio R_f/n^6 > 1 that is dominated by its last term, the leaves (R_f^(log_n W) = W^(log_n R_f) = O(W^sigma)); precisely T(W) = O(W^(log_n R_f)) = O(W^sigma) because the per-level cost +increases geometrically toward the leaves (ledger check `rect_leaf_domination`: sigma - (k+1) = sigma - 6 > 0; if R_f = n^6 exactly there is only a logarithmic factor, absorbed by the strict slack). Equivalently the ratio of the additions' cost +at the top, W^6, to the leaf cost W^sigma is W^(6-sigma) < 1. Per step of the band solve two such products are needed, working set W * n^5 W^5 * P bits (polynomial). The additions are complete block scans: O(W^6) entries per level, +O(W^6) per node up to logarithms, as in the square-case tape accounting of PR1673/PR1699 (depth-first scheduling, fixed block permutations, tape repositioning). + +## 3. Existence of an exact rectangular format and precision + +Let omega(1,1,5) < sigma0 := 6.156708 (cited). By definition of omega(1,1,5) (the exponent of the arithmetic complexity of ; arithmetic complexity and tensor rank have the same exponent up to the constant bilinearisation factor +that is absorbed by the strict slack), for every sigma > sigma0 there are fixed integers n >= 2 and R_f with an exact bilinear formula of rank R_f for and R_f < n^sigma: choose s with sigma0 < s < sigma, a constant C with +rank <= C n^s for all n, and n large (the cited analysis defines omega through the asymptotic rank, and the infimum is over exact formulas for tensor powers). This uses strict slack and never asserts that the infimum is attained by a finite formula. +The rank equations are finitely many polynomial equations over Z. A complex solution can be replaced by one over the algebraic closure of Q: otherwise, by the weak Nullstellensatz over Qbar, 1 would lie in the ideal, giving the same contradiction over C. +Enumerate finite algebraic tuples with isolating descriptions, test the equations exactly, dovetailing over (n, R_f); the search halts by the existence statement and its cost and description are fixed independently of the growing input +(the argument of PR1699/PR1673's matrix lemma with the format in place of ; no finite scheme is claimed as generated here). +Recursion: T(W) <= R_f T(W/n) + O(W^6), with W padded to the next power of n (a constant-factor change of the dimension; the batch width is the padded W^5, so both W and R are powers of n and no axis of the tensor is padded). +Precision: approximate each fixed algebraic coefficient once by precision-doubling Newton iteration, O(M(P')) bit operations for P' bits. The absolute roundoff envelope of the fixed bilinear recursion obeys E_j <= K E_{j-1} + K n^(6(j-1)) +with K fixed by the chosen formula (maximum L1 column sums of the encoders and decoder times the base-case size), hence E_j <= K^j (...) = W^(log_n K) poly(W), polynomial in W. Input magnitudes at most 2^h need fractional precision +P + 2h + O(log W) for absolute error 2^-P; intermediate magnitudes are W^O(1) max(1, 2^(2h)); the total record width is O(P + h + log W) = O(p) at the retained boundary interface (P, h = O(p)), and a larger fixed guard constant absorbs the formula-dependent K. +The bound is absolute, including cancellation. + +## 4. Choice of sigma, controls + +sigma = 6.1567081 is strictly above the published bound 6.156708 (margin 1e-7, costs about 1.4e-13 in kappa), so tau_eff = 6.1567081 - 4 = 2.1567081. Ledger checks: `rect_exponent_above_published_bound` (sigma - 6.156708 > 0), +`rect_leaf_domination` (sigma > k + 1 = 6), `rect_effective_exponent_above_two` (tau_eff > 2, the trivial output-size bound) and `rect_effective_exponent_below_five_halves`. Rejecting controls: sigma = 6.1567079 (below the published bound) +and sigma = 6.156708 (equal to it) must fail `rect_exponent_above_published_bound`; both are asserted to fail. + +## 5. Consequence for the ledger + +c = (3 - tau_eff)/2 = 0.42164595. The boundary row still binds with main = fused Gaussian (xi = epsilon - c + kappa); kappa = 0.0016781238306..., +1.312e-7 over the k = 2 package (pkg5) and +3.04e-7 over the square tau = 2.372864 package (pkg4). +At the new balance epsilon = 0.99903, xi = 0.5791, beta = 0.4209; cap xi > 1/2 holds with margin 0.0791; union saving xi + 1/2 - epsilon = 0.080 (the old g-factor union row would be -0.341); +group scans 0.422; support/key/membership rows about 0.579; union tag work 1 - 2 beta = 0.158. The unproven omega_C <= 9/4 premise is not used. + +## 6. What is not proved + +The rectangular exponent is a published theorem used as a cited input. The existence of the exact format and the stability constants rest on the arguments above and on PR1673's inherited all-size hypotheses +(compiler, precision/tape, exact recovery, analytic transfer); a reviewer should in particular check the claim that the number of right-hand sides in an axis substream is superpolynomial in the regime used +(spectator payload L^g) and the per-column normalisation in section 1. diff --git a/research/selective-borrowing-outer-retune/outer/RETUNED-SLACKS.md b/research/selective-borrowing-outer-retune/outer/RETUNED-SLACKS.md new file mode 100644 index 0000000000..898b4b1607 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/RETUNED-SLACKS.md @@ -0,0 +1,27 @@ +# Strict slacks of the PR1673 capped phase-cell fusion outer in this package + +Construction, proof and every formula are Vlado Vojdanovski's PR1673 (vendored verbatim under `vendor/pr1673`); the classical-matrix boundary +constant c = 3/10 is Chafik Boukhalfa's PR1699 specialisation of Francois Le Gall's published exponent (`CLASSICAL-BOUNDARY.md`). The following +constants are free in PR1673's own text and audits (each only has to be strictly positive) and are set as follows: + +| constant | PR1673 | PR1699 | here | why free | +|---|---|---|---|---| +| b (complex saving) | 17-decimal truncation | endpoint (22 digits) - 1e-30 | certified endpoint - 1e-60080 | must lie strictly inside the certified supplier endpoint | +| q gap (q = b - gap) | 1e-10 | 1e-30 | 1e-60080 | q < a_eff < b strictly | +| backoff after truncation | 1e-11 | 1e-42 | 1e-60022 | strict gap absorbs polylog factors | +| layout slack s0 | 1e-5 | 1e-30 | 1e-60080 | positive layout slack (carried by the boundary row in pkg4) | +| r0 | 1e-6 | 1e-30 | 1e-60080 | only eventual u = alpha^2 <= p^r0 | +| zeta | 1e-8 | 1e-32 | 1e-60082 | r + zeta > 0 gives g a B = o(p) | +| stop exponent | 1e-9 | 1e-36 | 1e-60086 | any t_stop > 0 with (1-t)b > a_eff, q < (1-t)b | +| decimal floor | 1e-18 | 1e-48 | 1e-60046 | rounding grid only | + +The exact enclosure of the native moment sum uses a 2^-199767 grid (directed-rounding integer fixed point, atanh/exponential term counts derived from the grid in check_ledger.py) so that moment(b) < 1 and +moment(endpoint + 1e-60000) > 1 are certified (the supplier endpoint is certified to 1e-60000 by the repaired fine engines). kappa = 0.0016776417313433545642517275958639984231708545722735432722117400060454625429077941422798526440584702 +(exact rational in `LEDGER.json`), above PR1699's exact kappa by 6.69e-25. + + +**Fine-prime package.** epsilon = 1 - DELTA_E with DELTA_E = 1.0009765625e-60080 (any DELTA_E > r0 = 1e-60080 is admissible, outer/FINE-PADDING-COMPOSITION.md), beta = 1 - 3 DELTA_E; all other slacks unchanged. + +**Fixed-grid maximum.** The free gaps above are the point of PR2180 / PR2161 (dataisfire; stacking on this series' PR2044, PR1786 chafreaky / Chafik Boukhalfa and PR1673 Vlado Vojdanovski): r0 = 1e-60080, zeta = r0/100, layout slack = gap = r0, +stop = r0/10^6, b = C - r0, q = C - 2 r0, epsilon = 1 - (1025/1024) r0, beta = 1 - 3 (1025/1024) r0; the certified endpoint C, the floor grid F = 10^60046 and the backoff B = 10^-60022 are unchanged. +The ledger kappa is then exactly C - 1/F - B, the maximum of the declared floor-grid/backoff family (`outer/PROOF-GRID-MAX.md`). This is a statement about this grid and backoff only, not about other suppliers, backoffs or finer grids. diff --git a/research/selective-borrowing-outer-retune/outer/STRUCTURED-BOUNDARY.md b/research/selective-borrowing-outer-retune/outer/STRUCTURED-BOUNDARY.md new file mode 100644 index 0000000000..311751158d --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/STRUCTURED-BOUNDARY.md @@ -0,0 +1,103 @@ +# Structured boundary solves (new in this package) + +> **Package note.** This construction is retained as an alternative but is **not used** by this package's kappa: with fine source-prime padding the boundary row is not binding (xi = 0.908 < epsilon), and the explicit Strassen boundary of PR1786 is used (outer/FINE-PADDING-COMPOSITION.md). The numbers in sec. 7 below belong to the earlier package. + +**Status.** This construction is a new contribution of this package. It builds on Vlado Vojdanovski's PR1673 (capped phase-cell fusion: the interior/boundary factorisation, the no-pivot boundary solver, the audited +conditioning bounds BILATERAL-LDU.md, FORWARD-SMALL-A-AUDIT.md, LOG-A-PHASE-SCHUR-AUDIT.md, PHASE-LAYOUT-AUDIT.md), on the radix-distributed union schedule (pkg4, outer/UNION-SCHEDULE.md) and on the classical structured-matrix +theory of Gohberg and Semencul (inversion of Toeplitz matrices, 1972), Kailath, Kung and Morf (displacement rank, 1979), Bitmead and Anderson (1980) and Pan (structured matrices and polynomial computations). It replaces the dense/fast-rectangular +products of the boundary algebra (pkg1 to pkg6: W x W blocks against W..W^k right-hand sides at cost W^tau) by applications of generator representations of the blocks: O(r W log W) ring operations per right-hand-side column, i.e. polylog(p) per entry. +The boundary row then has exponent epsilon - xi + 1/2 + o(1) (tau_eff = 2 + delta for every fixed delta > 0; tau_eff = 2 itself is not claimed). All inherited all-size hypotheses of PR1673 remain. The package pkg6 (rectangular k = 5, tau_eff = 2.1567081) is the fallback. + +## 1. Where the structure comes from + +Within one phase cell, with the notation of the note and of BILATERAL-LDU.md (rho = 1 + theta, q_i = floor(rho i + 1/2) = i + k on the cell's stretch of constant offset k, beta_j = rho j - q_j = theta j - k), +G_ij = exp(-a[(rho j - q_i)^2 - beta_j^2]). Expanding with rho j - q_i = (j - i) + beta_j gives the exponent (j-i)^2 + 2(j-i) beta_j = rho (j-i)^2 + theta (j^2 - i^2) - 2k(j - i), hence **exactly** + + G = D_r T D_c, T_ij = exp(-a rho (j-i)^2) (Toeplitz), (D_r)_i = exp(a theta i^2 - 2 a k i), (D_c)_j = exp(-a theta j^2 + 2 a k j). + +(The same factorisation, with a, rho and the offsets of the forward map, holds for the broad forward operator H of FORWARD-SMALL-A-AUDIT.md, which is itself a symmetric Toeplitz matrix after the anchored diagonal gauge, and for the augmented Haug = E H C + (I - E C), +which is permutation-similar to H (+) I.) The truncation to bandwidth W (rows farther than W are below 2^-P and are zero in the retained rounded matrix) keeps T banded Toeplitz. Cells are separated by physical cuts, which the note already makes boundary coordinates, so no block of the +boundary system couples two cells; the cell-wise offset k only enters D_r, D_c. + +**Diagonal cancellation.** In the interior/boundary decomposition the Schur complement S = D - C A^-1 B and every step of the block LU are built from products K_{ab} K_{bb}^-1 K_{bc} with K = D_r T D_c; +D_c(b) D_c(b)^-1 and D_r(b)^-1 D_r(b) cancel, so in the coordinates Y = D_r^-1 X D_c^-1 *every block of S, every LU pivot and every inverse pivot is the corresponding block of the Schur complement of the plain Toeplitz matrix T* (checked exactly in struct_check.py). The diagonals are only applied at +the input and output of a boundary solve, as the same anchored ratios (relative to the block anchor) as the capped gauge of CAPPED-GAUGE-AUDIT.md: their range over a block of the line, including consecutive boundary segments, is O(aB) with the linear (old) gauge of CAPPED-GAUGE-AUDIT.md, and O(a theta B'^2) = O(a B^2/d) (B' <= B + 2W) in the fractional-shift representation of GAUGEFREE-INTERIORS.md (chirp only), inside the group guard g a B^2/d = o(p) used by this package (beta + 2 xi - eps < 1). No new gauge is introduced. In the shifted representation the pivot bounds are those of GAUGEFREE-INTERIORS.md Corollary 4.3 (P_j^-1 is a principal block of the inverse of the accretive contiguous section T[J]); the displacement-rank statements are unchanged (they hold for any Toeplitz T). + +## 2. Displacement ranks (the algebraic statement) + +Let W x W be the segment size and Z the W x W down-shift. The displacement of a matrix M with row and column index sets that are intervals (shifts Z of the corresponding sizes) is Delta(M) = Z M - M Z (Sylvester) or M - Z M Z^T (Stein); the two ranks differ by at most 2 (Kailath-Sayed). +Facts for a banded Toeplitz T and interval index sets: +(a) every rectangular block of T has displacement rank <= 2 (the displacement lives in the first row and last column); +(b) for invertible Toeplitz A, Delta(A^-1) = -A^-1 Delta(A) A^-1 has rank <= 2 (Gohberg-Semencul); principal and corner blocks of such a matrix have rank <= 2 + 2; +(c) Delta(XY) = Delta(X) Y + X Delta(Y) with the middle shift matched, so ranks add under products; ranks do not increase under sums beyond the sum; Delta(M^-1) = -M^-1 Delta(M) M^-1, so M and M^-1 have equal rank; +(d) (Kailath) the Schur complement of a leading block of a matrix with displacement rank r has displacement rank <= r (w.r.t. the shift of the remaining indices). +Consequences for the boundary J = G separated W-segments with interiors I_j between them (all interiors eliminated): +* coupling S_{j,j+1} = - T_{j,I_j} A_{I_j}^-1 T_{I_j,j+1}: rank <= 2 + 2 + 2 = 6 (a, b, c); S_{j+1,j} likewise; +* diagonal block S_{jj} = T_{jj} - T_{j,I_{j-1}} A^-1 T_{I_{j-1},j} - T_{j,I_j} A^-1 T_{I_j,j}: rank <= 2 + 6 + 6 = 14; +* the LU pivot P_j = S_{jj} - S_{j,j-1} P_{j-1}^-1 S_{j-1,j} is the Schur complement of T onto segment j after eliminating all indices to its left (a leading block of T: rank <= 2 by (d)) minus the single right-interior correction T_{j,I_j} A^-1 T_{I_j,j} + (rank <= 6; indices farther to the right do not couple to segment j, in T or in its left Schur complement, because the band is W and the separators have length >= W): **rank(P_j) <= 8 uniformly in j and G**, and rank(P_j^-1) = rank(P_j). +A *global* displacement operator on the concatenated J has rank growing like 2G (IdeaLiterature's screen); the claim here is block-wise, with one shift per segment, which is exactly what a block-tridiagonal LU needs. The constants 14 and 8 are proved bounds; the observed ranks are <= 4 (struct_check.py, +and IdeaLiterature's 70-110-digit screens of the real chirp-weighted blocks: coupling 2, diagonal and pivot 2-4, flat in W and in G). + +## 3. Generator representation + +A block M with Stein displacement M - Z M Z^T = sum_{k<=r} g_k h_k^T is represented by (g_k, h_k) and recovered by the Kailath-Kung-Morf formula **M = sum_k L(g_k) L(h_k)^T**, L(g) lower-triangular Toeplitz with first column g +(rectangular off-diagonal blocks use the same formula with the matched shifts). Generators are computed **offline, from generators, never from the dense blocks**: the leading Schur complements of the banded Toeplitz T are propagated by the generalised Schur algorithm (r = 2 generators, +O(r W) work per eliminated index, or the superfast O(r W log^2 W) variant of Bitmead-Anderson and Pan per segment); the interior corner blocks T_{j,I} A_I^-1 T_{I,j'} come from the Gohberg-Semencul generators of A_I^-1 (first and last columns of the inverse of the interior Toeplitz block); +the pivot and inverse-pivot generators are then obtained by the rank-<= 8 recompression of the sum (generators of the Schur piece and of the correction), an exact operation on ranks (QR/SVD at working precision P + poly bits). This is the page regeneration the audit already charges +(polynomial in p per complete microbox and axis, amortised by the superpolynomial payload); the generator pages replace the dense W x W factor pages. Balanced generators from the SVD of the displacement (g_k = sqrt(sigma_k) u_k, h_k = sqrt(sigma_k) v_k) satisfy |g_k||h_k| = sigma_k <= ||Delta(M)|| <= 2||M||, +so each term of the KKM sum has norm <= W ||Delta(M)|| (|L(g)|_2 <= |g|_1 <= sqrt(W)|g|_2) and the sum has norm <= 2 r W ||M|| = poly(W) ||M||: no hidden cancellation beyond a polynomial factor. + +## 4. The solve on the tape + +For one boundary band the block-tridiagonal system S x = y (stage diag(I, S^-1) of the audit; the C and B stages are products with the banded Toeplitz coupling blocks T_{J,E}, themselves rank-2 Toeplitz blocks) is solved by block forward and backward substitution with the audited no-pivot LU: + z_j = P_j^-1 (y_j - S_{j,j-1} z_{j-1}), x_j = z_j - P_j^-1 S_{j,j+1} x_{j+1}. +Each operator application (S_{j,j±1}, P_j^-1 and the Toeplitz couplings) is r pairs of triangular-Toeplitz products L(g) (L(h)^T v), each a convolution of length W computed by FFT of length 2W at working precision: O(r W log W) ring operations of p-bit numbers per right-hand-side column. +Layout: the right-hand sides of an axis substream (the lines of the union schedule, UNION-SCHEDULE.md) are batched in the dyadic width of the audit; the batch matrix |E_f| x batch width is arranged so that the W-axis of every column is contiguous by the small-field transpose (O(log p) riffle passes), the FFT is a sequence of O(log W) butterfly +passes, each a scan with a bit-reversal permutation realised by O(log W) riffle passes: O(log^2 W) scans of the batch data, polylog p overhead. The generator page of a block (r W numbers) is scanned once per batch; with the batch width >= W this is a 1/W fraction of the batch data (as for the dense pages in the audit). +Cost: per band and per column O(r W log^2 W M(p)) bit operations, i.e. per entry O(r log^2 W M(p)) = (M(p)/p) polylog p * p; with the retained short multiplier M(p)/p = polylog p, the boundary arithmetic is **O(V (d/B) W polylog p)**, so the boundary exponent is epsilon - xi + 1/2 + o(1), i.e. tau_eff = 2 + delta (any fixed delta > 0 absorbs the polylog factors since W = p^(1/2) polylog p). +(The ledger uses delta = 1e-60080, the same convention as the other strict slacks.) + +## 5. Precision and stability at Cp bits (the regime: row gap about 1/(16p), poly(p) conditioning) + +The argument reuses the audited bounds and adds the factors introduced by the structured representation. +1. *Pivots and couplings.* By PR1673's no-pivot theorem (BILATERAL-LDU.md Sections 1-2: unit L and U with all entries and all inverse entries bounded by constants times exp(-a rho theta |i-j|), c = prod(1 - r^-m) > 0, ||N^-1||_inf < 16 theta^-2; and the narrow/broad row-gap bounds of LOG-A-PHASE-SCHUR-AUDIT.md and FORWARD-SMALL-A-AUDIT.md: + Schur row gap >= 1/(16 p) survives elimination), every P_j^-1 is a principal block of the inverse of a partially eliminated matrix and satisfies ||P_j^-1||, ||S_{j,j±1}||, ||P_j|| <= poly(p) uniformly in j, with the unit L^-1 / U^-1 bounded independently of G (this is the audited no-factor-equal-to-the-number-of-windows property). In the SPD (broad, Saug/4) case the principal-submatrix inverse has norm <= lambda_min(H)^-1 <= poly(p). +2. *Generators.* From section 3, balanced generators have norm <= sqrt(2 ||M||) = poly(p)^(1/2) and the KKM reconstruction has norm <= 2 r W ||M|| = poly(p). +3. *One application.* An FFT convolution of length 2W in fixed point with Cp-bit coefficients and a precomputed table of 2W roots of unity (computed once, offline, to P + poly bits by Newton iteration in O(M(P'))) has absolute error <= O(log W) 2^-P' times the product of the l1/l2 norms of its operands. So one application of an operator M to a vector v has + absolute error <= poly(p) ||v|| 2^-P' + (the error of the stored generators, <= 2^-P' poly(p)). The diagonal gauges D_r, D_c enter only through the anchored ratios of range O(aB) (section 1), which cost O(aB) = o(p) extra bits: the same guard as the interiors (chirp-only O(a B^2/d) per axis in this package; O(aB) in the earlier linear-gauge version); no exponential growth beyond it. +4. *Accumulation.* The forward and backward substitutions are the unit L^-1 and U^-1 of the audited LU followed by diagonal factors P_j^-1; errors of different steps enter linearly through the unit factors whose inverse is bounded independently of G, so the total absolute error is <= poly(p) 2^-P' (a sum over at most poly(p) = L/B segments of poly(p) terms, with the audited exponential decay of the L^-1, U^-1 entries + making the dependence on G at most polynomial). Hence P' = P + O(log p) + o(p) (the guard), i.e. O(p) bits suffice, the same record width as for the dense algorithm. +5. *Offline generator accuracy.* Generators are generated at working precision P + poly(p) bits in polynomial time; their conditioning is that of the blocks themselves (the generalised Schur algorithm is applied to the banded Toeplitz T with strictly diagonally dominant / SPD structure whose leading Schur complements are bounded below by the row gap), so no loss beyond poly(p) factors occurs. +The one point where structured algorithms are known to be unstable in general (indefinite or nearly singular leading minors) is excluded here by the audited uniform lower bounds on the pivots (item 1); this is where an independent reviewer should look first. + +## 5b. Cleaner conditioning argument, rounding convention, anchors and pages (added after independent review) + +* **Conditioning without the no-pivot theorem.** In the coordinates Y = D_r^-1 X D_c^-1 every block, pivot and inverse pivot is a block or a Schur complement of the plain symmetric positive definite Toeplitz matrix T (a Gaussian symbol is positive for every a; the narrow case has cond(T) = 1 + O(e^-a)). + By interlacing and monotonicity of Schur complements, every principal submatrix and every Schur complement of T has its spectrum inside [lambda_min(T), lambda_max(T)], hence every leading minor and every pivot P_j (and P_j^-1) has condition number at most cond(T), and cond(T) <= (1 + sqrt(pi/gamma))/(sqrt(pi/gamma) e^(-pi^2/(4 gamma))) = O(e^(a/4)) = O(p) + with the eigenvalue bounds of FORWARD-SMALL-A-AUDIT.md (the same constants as the broad forward audit). The Bunch-type danger of structured solvers (indefinite or nearly singular leading minors) cannot occur for SPD T; the Schur/Levinson recursion and the Gohberg-Semencul application on an SPD Toeplitz matrix are weakly stable, losing O(log cond) = O(log p) bits. + (This replaces the appeal to the row gap 1/(16 p) in section 5 item 1 for the SPD blocks; the row-gap bounds remain the justification for the non-symmetric narrow blocks. Independent numerical evidence, LeverRank, 80-160 digits, alpha = 4, a = 50.3, rho = 1 + 1/6000: pivot condition numbers saturate at cond(T) ~ 1.4e5 for W = 16, 32, 64; balanced-generator norm ratio sigma_1(grad L^-1)/||L^-1|| 0.14-0.27 (no growth); generator-to-inverse rounding amplification <= 1; + Levinson/Gohberg-Semencul error about 2^9.5 u flat in n; a perturbation of the first pivot is damped along the chain, so accumulation is independent of G.) +* **Rounding convention (required).** The retained rounded matrix must be rounded *factorwise*: K^ = D^_r T^ D^_c with T^ exactly Toeplitz (symmetric rounding of exp(-a rho x^2) by |i-j|) and D^_r, D^_c the rounded gauge weights. This is the structure-preserving rounding of FORWARD-SMALL-A-AUDIT.md and the Toeplitz-similar cell matrix of LOG-A-PHASE-SCHUR-AUDIT.md. Entrywise rounding of K would destroy the Toeplitz structure (this is the dense non-Toeplitz control of section 6: the displacement rank jumps to W). +* **Anchors.** The diagonal conversions between X and Y coordinates use segment-local anchors. The inter-segment constant ratios D_r(m_j)/D_c(m_{j-1}) (about e^{O(aB)}) are stored inside the coupling pages and cost O(aB) = o(p) guard bits per page; no range over the whole line is ever formed. +* **Pages.** In the interior of a line the pivots are identical (observed), so each cell needs O(1) distinct generator pages (only the cell-end blocks differ): page regeneration is cheap. A batch width W suffices (no W^k batching is needed in this package). + +## 6. Executable check (`outer/struct_check.py`) + +On the exact rational model K = D_r T D_c with a random (non-symmetric) banded Toeplitz T, W = 3..5, G = 3..6 segments, interiors of length 3W: (1) the exact diagonal cancellation S_Y = Schur(T); (2) the Stein displacement ranks of every segment block, coupling block, LU pivot and inverse pivot (all <= 4, flat in the segment index and in G); +(3) exact reconstruction of every block from its rank factorisation by the KKM formula; (4) the block forward/back substitution performed only with generator representations equals the dense solve exactly; (5) controls: a dense non-Toeplitz perturbation of the band raises the pivot and diagonal ranks to the full W (here 6 > 4), and a perturbed pivot is not reproduced by the original generators. +This checks the algebra exactly at small W; it does not check the analytic stability argument of section 5, which is an argument, nor does it replace a check on the real chirp-weighted blocks at 70-110 digits (IdeaLiterature's screens: scratch/IdeaLit/real.py, real2.py). + +Scope of the real-matrix screen (IdeaLiterature, disclosed): block ranks 2-4 were observed for W <= 16 and G <= 8 segments only. In that toy the pivot condition number grows with W (about 3.5 at W = 8 and 1.2e3 at W = 16) because the toy takes a proportional to 1/W^2. +In the regime of the note a = pi alpha^2 >= 1 is fixed independently of W (alpha = ceil(sqrt((4/pi) log 16p))), theta >= 1/(4d), and BILATERAL-LDU.md bounds the periodic inverse by ||N^-1||_inf < 16 theta^-2 and the pivots/factors uniformly (items of section 5), so the conditioning is polynomial in p; the toy's growth with W is therefore not the real regime, but it is exactly the point that is *not* verified numerically here. + +## 7. Ledger consequence + +tau_eff = 2 + delta (delta = 1e-60080): c = (3 - tau_eff)/2 = 1/2 - delta/2. Balance: main = fused Gaussian = boundary - slack with xi = epsilon - c + kappa; the union row xi + 1/2 - epsilon has the same exponent as the boundary row (both epsilon + 1/2 - xi), so both bind at kappa + 1e-60080 together with main and the Gaussian row; +with the chirp-only guard of GAUGEFREE-INTERIORS.md: xi = 0.50167, xi - 1/2 = 0.00167 > 0 (cap holds), beta = 0.99665, union tag work per visit 1 - beta = 0.00335 > 0. kappa = 0.00167974550880...; +1.40e-6 over the structured-boundary balance with the linear guard (kappa 0.0016783441717...; that was +2.20e-7 over k = 5, tau_eff = 2.1567081). + +## 8. What is not proved + +* The constants c0 = 14, 8 follow from the displacement calculus (a)-(d) above; (d) and the Gohberg-Semencul statement (b) are classical theorems used as inputs; the corner-block increment (+2) and the application to the real chirp-weighted blocks (Haug, Saug, narrow and broad pairs) rest on the exact factorisation of section 1. +* Section 5 is an error argument built from the audited bounds, not a verified stability theorem for structured solvers. The real-regime numerical check (row gap 1/(16 p), p large) has not been run. +* Everything inherited from PR1673 (all-size compiler, precision/tape, exact recovery, analytic transfer) remains an assumption. diff --git a/research/selective-borrowing-outer-retune/outer/UNION-SCHEDULE.md b/research/selective-borrowing-outer-retune/outer/UNION-SCHEDULE.md new file mode 100644 index 0000000000..56bf6993c9 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/UNION-SCHEDULE.md @@ -0,0 +1,132 @@ +# Radix-distributed sparse-union schedule (new in this package) + +**Status.** This schedule is a new contribution of this package. It is not part of PR1673 (Vlado Vojdanovski, whose capped phase-cell fusion +construction and PHASE-LAYOUT-AUDIT.md this document modifies and relies on), nor of PR1699 (Chafik Boukhalfa, Le Gall's exponent). It replaces +one step of PR1673's sparse-union window, the per-axis scan of U, and is conditional on every other hypothesis of PR1673. Its executable finite +check is `outer/sched_test.py`; the arithmetic consequence is `outer/check_ledger.py`. An independent review is required before release. + +## 0. Model and what is replaced + +Machine model: PR1673's fixed-tape model, i.e. a multitape Turing machine with a fixed number of tapes; the cost of an operation is the total head +travel; a record is a p-bit coefficient record with a header of O(g log L) bits (g=p^beta, L=poly(p) the microbox side, beta<1, so the header is o(p) +and moving it multiplies the cost by 1+o(1)). V is the number of records of the microbox (including spectators). All notation (E_i, J_i, T_i, mu, +W, B, d, g, L, the windows) is that of PHASE-LAYOUT-AUDIT.md and the capped interiors of CAPPED-INTERIORS.md; mu = max_i |E_i|/L = O((W/B) polylog p). + +A sparse window applies the Kronecker product `Theta = (x)_{i=1..g} T_i` to the microbox tensor, where `T_i = I_{E_i^c} (+) T_i|E_i` (for the window +containing the C and S^-1 stages, T_i is the product of the two sparse factors of axis i; each is block diagonal in the same way). +The audit's implementation scans the union U = {x : some x_i in E_i} once per axis to copy U_i = E_i x [L]^(g-1): cost O(|U|) per axis, hence +`O(V [g^2 mu + g mu polylog p])` per window and the union row `eps + beta + 1/2 - xi`. Here the g-fold scan of U is removed. + +## 1. Flag sets and class decomposition + +For a record x put `F(x) = {i : x_i in E_i}` (the g membership flags the audit already computes in the single scan that forms U; stored as a g-bit vector +in the header). + +**(F1) invariance.** `T_i` changes a coordinate x_i only inside E_i and fixes E_i^c, and maps E_i to E_i. Hence `F(Theta x) = F(x)`: every T_j preserves every class +`C_F = {x : F(x) = F}`. + +**(F2) commutation.** Each T_i depends only on the axis-i coordinate (and on axis i's own coarse label/page, fixed per microbox), so T_i acts on tensor factor i alone; the T_i act on different tensor factors, hence commute, and on class C_F the operator Theta equals `(x)_{i in F} T_i|E_i` on the coordinates in F +(range E_i) tensored with the identity on the coordinates not in F (the T_j, j not in F, are the identity on E_j^c). + +**(F3) complete lines.** Fix F, f in F and the other coordinates z (all coordinates except x_f). The *E_f-line* of z is {x : x_f in E_f, other coordinates z}. Every point +of it has the flag set `F(z) union {f}` (flags of z-coordinates plus f), so the whole E_f-line lies in one class, and contains exactly |E_f| points. + +## 2. The schedule + +Let `f_r(F)` be the r-th smallest element of F. Define `S_r = {x : |F(x)| >= r}`, kept as a stream in lexicographic order of x (S_1 = U; S_{r+1} is S_r with +the records |F|=r deleted, so lexicographic order is inherited). + +Round r (r = 1,2,...,R, where R = max |F| <= g) applies to every record x of S_r the operator T_{f_r(F(x))} along axis f_r(F(x)), to complete E_f-lines (F3). +After the rounds, every record of class C_F has received T_i exactly once for each i in F (the f_r(F) enumerate F), in increasing axis order; +by (F2) this equals the action of Theta. Since lines of the same round and class are disjoint and independent (F2), applying T_f to each line separately equals +the action of `T_f|E_f (x) identity` on the class. Rounds use the values written by earlier rounds because records carry their current values in the stream. + +### 2.1 One round (cost accounting per step; N = |S_r|) + +1. **Tag.** For each record compute `t = f_r(F(x))` from its header bits: O(g) bit operations (scan for the r-th set bit). Worst case a record has |F| = g and takes g rounds, + so tag work per visit is O(g) bit operations, paid by the Theta(p) cost of that visit (Section 3, header and tag work): g log L = o(p) since beta < 1 (ledger check `union_tag_work_per_visit_fits_record`). +2. **Stable binary LSD radix distribution by tag.** For k = 0..ceil(log2 g)-1: scan the stream; records whose bit k of the tag is 0 go to output tape Z, bits 1 to tape O + (one input tape, two output tapes), then Z is copied followed by O back onto the work tape. Each pass costs <= 2N record moves; the bits of the pass are written (N bits) + to a bit tape together with the count n0 of zeros. *Stability* by induction on k: after pass k the stream is ordered by (bits k..0 of tag) with ties in the original order. + After all passes: ordered by tag, ties in lexicographic order; the stream splits into contiguous axis substreams, one per tag value f. + Inversion (used in step 5): replay the passes in reverse: copy the stream onto Z/O at the stored split point n0 (one pass), then read the bit tape and take the next record from Z or O accordingly. + Cost 3N per pass. Total O(N log g) = O(N log p). +3. **Structure of an axis substream (the key lemma).** Axis f substream = {x in S_r : f_r(F(x)) = f}. Membership depends on F(x) only, and F(x) = F(z) union {f} for x with x_f in E_f (all substream records + have x_f in E_f, because f is in F(x)); so it depends on z only, not on x_f. Therefore for each fixed prefix x_f} is the same for every x_f in E_f. In the stream (lexicographic order of (prefix, x_f, suffix)) the prefix block is the concatenation of + a = |E_f| rows (one per x_f in E_f, in increasing order), each row listing the same b = |Z'(prefix)| suffixes in the same order. No coordinate is invented: every needed point is present. + Block boundaries are detected by comparing the prefix fields of consecutive records (O(g log L) bits per comparison, o(p)); rows by the ordinal of x_f in E_f stored in the header + (computed with the flags: the rank of x_f in E_f costs O(polylog p) with the carving description) -- no counter of unknown length is needed. +4. **Variable-length lockstep transpose of every prefix block** (brings x_f to the line suffix). Let `a = |E_f| <= L = poly(p)`. All rows of a prefix block have the same length b within a tag (point 3), b unknown to the routine. Recursively halve the row index range into the upper part of ceil(a/2) rows and the lower part of floor(a/2) rows (the implementation uses chunk sizes h = floor(a/2) for the first child and a-h = ceil(a/2) for the second; either order works, the tree must be the same for every block and its inverse). + Level structure: a balanced binary tree of row ranges of depth ceil(log2 a) = O(log p), the same tree for every block. *Merge of a node*: its children's outputs `TA` (rows 0..h-1 in suffix-major order, + i.e. for each suffix the h records of A's rows) and `TB` (the a-h rows of B) are read in lockstep from two tapes, taking h records from TA, then a-h records from TB, alternately, until both end markers + coincide; the output is the node's suffix-major order `[row_0[j], ..., row_{a-1}[j]]_j`. The routine never needs the row length b: the chunk sizes h and a-h depend on a alone, and the end of the node's data is + the end marker (prefix-block boundary, or the stream end). A level is realised by two passes over the whole substream. Every record carries in its header the ordinal k of its source row x_f. The data of a node (a range [lo, lo+n) of ordinals) is a contiguous run of records of one prefix block whose ordinals lie in the range; its child A (ordinals [lo, lo+h)) comes first, immediately followed by child B (ordinals [lo+h, lo+n)), so the junction is the first record whose ordinal is >= lo+h, and the end of the node is the first following record whose ordinal is >= lo+n or whose prefix field differs (both read from headers; the tree is the same for every block and nodes appear in row order). Pass 1 copies the A-parts to tape M1 and the B-parts to tape M2, node by node. Pass 2 reads M1 and M2 in lockstep, h records from M1 then n-h from M2, node by node (a node's A- and B-parts end at the same node marker), writing the parent level. So a level costs O(N) record moves and the whole transpose O(N log a). After the transpose, each line (fixed prefix and suffix) occupies a consecutive group of a records with increasing x_f ordinal. +5. **Apply T_f on lines and batch.** Each consecutive group of a records is a complete E_f-line; T_f|E_f is applied exactly as in the audit's step 4: consecutive lines form right-hand sides, batched in dyadic width between W and 2W, + factor pages (regenerated once per microbox and axis from its coarse origins, polynomial in p, amortised by the superpolynomial payload) scanned once per batch, only the polynomial-width row field moved + (O(log p) times the boundary volume). This arithmetic is the *boundary-algebra row*, unchanged and not part of the union movement. +6. **Undo** the transpose (unzip levels), then undo the radix distribution (step 2 inverse): the processed records are back in lexicographic order of S_r. +7. **Split.** One scan: records with |F| = r are final for this window; push them (with the |S_r|-bit indicator keep_r = [|F| > r]) on the LIFO tape; the rest, S_{r+1}, is written to the alternate work tape. + +### 2.2 Wrap-up + +After the last round, pop the LIFO tape: for r = R,...,1 interleave `rest_r` with the already final stream S_{r+1} by keep_r (the audit's merge by indicator), producing the final S_1. +Finally merge S_1 into the microbox using the indicator bits of the single scan that formed U; identity records (F = empty) are never moved. +The two dense-stage scatters/regathers, the one scan forming U and the final merge are the audit's group scans: O(V polylog p) per window, in the group-scan row. + +### 2.3 Tape-head accounting (per round, and why rounds do not multiply the cost) + +Tapes: W1, W2 (alternating work tapes carrying S_r), Z, O (radix outputs), Tb (bit tape), STACK (LIFO), M1, M2 (merge/zip tapes), small header/counter tapes: a fixed number. +* W1 holds S_r written from cell 0; before round r the head is at cell 0 (rewind over the written length |S_r| only; cells beyond are stale and never read). +* S_{r+1} is written to W2 from cell 0, then the roles swap. Rewinding costs the written length, at most |S_r|. +* STACK is only appended to at its end and, in the wrap-up, read backwards from its end; the rest_r are consecutive, so popping costs the sum of |rest_r| <= sum |S_r|. No head ever travels over the whole of + the stack or of U per round. +* Z, O, Tb, M1, M2 are written from cell 0 and rewound over their written length (<= |S_r| records or bits). +Hence the head travel of round r is O(|S_r| p (log g + log a + 1)) and the total is O(p (log g + log L + 1) * sum_r |S_r|). The rounds are sequential, but their *number* does not multiply the cost: |S_r| shrinks and +nothing in round r touches data outside S_r. + +### 2.4 Volume identity + +`sum_r |S_r| = sum_x |F(x)| = sum_i |U_i| = sum_i |E_i| L^(g-1) * (spectators) <= g mu V.` (Each record is counted once per flag; the first equality is exchanging the order of summation.) + +## 3. New cost and row + +Per window: `O(V g mu polylog p)`; no factor g^2 and no per-axis scan of U. Per group, then over d/g groups: + +`O(V (d/g) g mu polylog p) = O(V d W polylog p / B)`, + +i.e. union-movement exponent `eps + 1/2 - xi` (saving `xi + 1/2 - eps`). Old audit row: `eps + beta + 1/2 - xi`. +The identity `sum|F| <= g|U|` shows the new cost is never larger than the old one. + +*Sparsity is not used.* In the regime of this package the expected number of flags g*mu = p^(beta+1/2-xi) is larger than 1 (xi - beta - 1/2 < 0), so U is essentially the whole microbox, and the audit's ledger check +`sparse_union_fraction_decays` is not used. The cost is the total number of record-flag incidences, not |U|. +Header and tag work: the flag header has g bits and g log L bits of coordinates (o(p) since beta < 1); tag work is at most O(g) per visit and a record is visited at most |F| <= g times: each visit costs Theta(p polylog p) anyway, so the total over the sum_r |S_r| <= g mu V visits is O(g mu V (p polylog p + g log L)) = O(g mu V p polylog p) as soon as g log L = o(p), i.e. beta < 1. (An earlier version of this note charged O(g^2) per record against o(p) and required 2 beta < 1; that per-record total is not needed, see outer/GAUGEFREE-INTERIORS.md section 7.) + +## 4. Consequence for the ledger + +Rows: main `eps q`; fused Gaussian `1 - eps + q beta`; boundary `c + xi - eps` with `c = (3 - tau)/2`, `tau_eff = 2 + 1e-60080` (structured boundary solves, outer/STRUCTURED-BOUNDARY.md; this document's schedule is independent of the boundary exponent); +union `xi + 1/2 - eps`; guard `beta = 1 - xi - r - zeta` (linear gauge; in this package the guard is the chirp-only `beta = 1 - (2 xi - eps) - r - zeta`, see Section 5b); `xi > 1/2`. Balance set: main = fused Gaussian = boundary - slack, which gives +`eps = (1 + q + q (c - s - r - zeta))/(1 + q)^2`, `xi = eps (1 + q) - c + s`. The union row is far above kappa at this point (saving about 0.080 here). +Rejecting controls in the ledger: reinstating the old row `1 - eps - beta - 1/2 + xi` at the new point fails (it is about -0.125); the full-key sort row fails; the four inherited controls fail. + +## 5. Executable check (`outer/sched_test.py`, and the independently coded `outer/sched_review.py` by LeverRank) + +The script runs the schedule above on exact integer tensors (g = 3..6 axes, L = 5..8, random E_i and random integer T_i|E_i): tagging, the stable LSD radix distribution with its inversion, per-prefix-block variable-length lockstep zip +transposes with a check that every extracted group of a records is a complete E_f-line with consecutive ordinals and one common z, application of T_f, unzip, undo, the keep/rest split, the LIFO pop and the merge by indicator bits, +and compares with the full tensor product `(x) T_i` computed by tensor contraction: exact equality. It also checks the volume identity `streamed = sum|F| = sum_i |U_i|`, `sum|F| <= g|U|` (), +that the number of record moves is within `3 * sum|F| * (ceil(log2 g) + 2 ceil(log2 a) + 4)`, and a tamper control (skipping the second round) fails. + +## 5b. Rows that depend on beta, re-verified at the balance of this package + +At the fine-prime-padding point (outer/FINE-PADDING-COMPOSITION.md; Strassen boundary c = 3/32, theta ~ p^-2): epsilon = 1 - 1.0009765625e-60080, beta = 1 - 3.0029296875e-60080, xi = 0.90792965...; every row is regenerated exactly by `check_ledger.py` and re-derived by `independent_check.py` (all strictly above kappa or strictly positive): +group scans 1-eps+beta = 1 - 2.001953125e-60080 (saving far above kappa); union xi+1/2-eps = 0.4079 (boundary = main + 1e-60080 binds; the union is no longer binding; the old row 1-eps-beta-1/2+xi = -0.5921 fails: rejecting control, so the radix schedule is still essential); +cap xi-1/2 = 0.4079 > 0 and eps-xi > 0.092; group positive and below dimension (eps-beta = 2.001953125e-60080 > 0); support router span 1-beta = 3.0029296875e-60080 > 0 (the supplier's total moved support O(g polylog p) = o(p) holds eventually, with exponent margin 3.0029296875e-60080); +gauge/guard 1-(2xi-2)-r-beta = 0.184 (fine theta; the linear guard 1-xi-r-beta is negative: rejecting control); group key fits the record (eps-beta = 2.001953125e-60080); membership arithmetic 1-beta-2(1-eps) = 1.0009765625e-60080; union tag work per visit 1-beta = 3.0029296875e-60080 (the old test 1-2beta is negative and not used); union header 1-beta; core-face corrections eps+beta+48-128+2 < 0. +The ledger row `sparse_union_fraction_decays` (xi-beta-1/2 > 0) stays deleted: it is false (g mu = p^(beta+1/2-xi) > 1) and the schedule does not use it. + +## 6. What this does and does not establish + +Established here: the data-movement schedule, its correctness (class invariance, commutation, complete lines, stability of the radix passes, the lockstep transpose, LIFO interleave) and its tape-head cost O(V g mu polylog p) per window, assuming +the audit's carving (|E_i| and mu), the retained polynomial-width row-field movement for batched boundary arithmetic, and the retained dense/gauge/precision lemmas of PR1673. +Not established: nothing in this document changes the arithmetic, the Gaussian conditioning, the supplier, the boundary matrix exponent or the all-size hypotheses of PR1673; those remain inherited assumptions. +The step "variable block length" of the transpose (item 4) and the head accounting (2.3) are the places that most need an independent review. diff --git a/research/selective-borrowing-outer-retune/outer/check_accretive_scale.py b/research/selective-borrowing-outer-retune/outer/check_accretive_scale.py new file mode 100644 index 0000000000..e88786118b --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/check_accretive_scale.py @@ -0,0 +1,87 @@ +#!/usr/bin/env python3 +"""Exact checks for outer/ACCRETIVE-SCALE.md (scale qualification of GAUGEFREE-INTERIORS Theorem 4.1, per the PR1758 review +https://github.com/CrocSwap/integer-mult-bounds/pull/1758#issuecomment-6203527654). Fractions and SymPy only, no floats. + + A the review's counterexample N = 2, H = tI: Lambda/mu = 1 for every t, but max(||H||, ||H^-1||) = max(t, 1/t) is unbounded, + while the absolute-scale guard 2 + Lambda + 1/mu dominates it. + B unit diagonal (Lambda >= 1, mu <= 1): 2 Lambda/mu - Lambda - 1/mu >= 0 (polynomial with nonnegative coefficients). + C descendants (Lambda' = Lambda + Lambda^2/mu, inverse 1/mu, shears Lambda/mu) on unit diagonal: + Lambda' + 1/mu <= 3 (Lambda/mu)^2 and 2 + Lambda' + 1/mu <= 4 (2 + Lambda/mu)^2. + D near-unit diagonal fallback (|h0 - 1| <= 1/2, so |h0| >= 1/2, Re h0 <= 3/2): Lambda + 1/mu <= (7/2) Lambda/mu. + E the applied kernels have diagonal exactly one: G'_s(0) = 1 and the narrow N_ii = 1 symbolically; on exact rational + truncated/rounded Toeplitz sections (diagonal 1, as the circuit applies them) e0* H e0 = 1 (so mu <= 1) and + ||H e0||^2 >= 1 (so Lambda >= 1), and the Hermitian part is positive definite by exact LDL^T. +""" +from fractions import Fraction as Q +import sympy as sp + +checks = 0 +def ok(c, why): + global checks + if not c: raise SystemExit('FAIL: ' + why) + checks += 1 + +# A. counterexample +for k in range(-40, 41, 5): + t = Q(2) ** k + Lam, mu = t, t # H = tI: ||H|| = t, Hermitian part = tI + ok(Lam / mu == 1, 'A ratio') + worst = max(t, 1 / t) # max(||H||, ||H^-1||) + ok(2 + Lam + 1 / mu >= worst, 'A absolute guard dominates') +ok(max(Q(2) ** 40, Q(2) ** -40) == Q(2) ** 40, 'A unbounded at fixed ratio') + +# B, C, D symbolic: Lambda = L0 (1 + a), mu = m0 / (1 + b), a, b >= 0 +a, b = sp.symbols('a b', nonnegative=True) +def nonneg_poly(expr, why): + num, den = sp.fraction(sp.together(sp.expand(expr))) + pn, pd = sp.Poly(sp.expand(num), a, b), sp.Poly(sp.expand(den), a, b) + ok(all(c >= 0 for c in pn.coeffs()) and all(c >= 0 for c in pd.coeffs()), why) +Lam, mu = 1 + a, 1 / (1 + b) +r = Lam / mu +nonneg_poly(2 * r - Lam - 1 / mu, 'B unit diagonal: Lambda + 1/mu <= 2 Lambda/mu') +Lp = Lam + Lam ** 2 / mu +nonneg_poly(3 * r ** 2 - Lp - 1 / mu, 'C descendant: Lambda prime + 1/mu <= 3 (Lambda/mu)^2') +nonneg_poly(4 * (2 + r) ** 2 - (2 + Lp + 1 / mu), 'C descendant log form') +nonneg_poly(r - Lam / 1, 'C shear Lambda/mu >= Lambda') # shears <= Lambda/mu, inverse 1/mu <= Lambda/mu +nonneg_poly(r - 1 / mu, 'C inverse 1/mu <= Lambda/mu') +LamD, muD = sp.Rational(1, 2) * (1 + a), sp.Rational(3, 2) / (1 + b) +nonneg_poly(sp.Rational(7, 2) * LamD / muD - LamD - 1 / muD, 'D near-unit diagonal: Lambda + 1/mu <= 7/2 Lambda/mu') + +# E. the applied kernels +g, s, u, aa, b0, th, t = sp.symbols('gamma s u a beta0 theta t', real=True) +Gp = sp.exp(-g * u ** 2 + 2 * g * s * u) +ok(sp.simplify(Gp.subs(u, 0) - 1) == 0, "E G'_s(0) = 1") +ok(sp.simplify(sp.exp(g * s ** 2) * sp.exp(-g * (u - s) ** 2) - Gp) == 0, "E G'_s normalisation identity") +# narrow block N_{i,i+h} = exp(-a[(rho h + beta_i)^2 - beta_{i+h}^2]) at h = 0 +h = sp.symbols('h', real=True) +bi = b0 + th * t +N = sp.exp(-aa * ((1 + th) * h + bi) ** 2 + aa * (bi + th * h) ** 2) +ok(sp.simplify(N.subs(h, 0) - 1) == 0, 'E narrow N_ii = 1') + +def ldl_pd(M): + n = len(M); A = [row[:] for row in M] + for k in range(n): + if A[k][k] <= 0: return False + for i in range(k + 1, n): + f = A[i][k] / A[k][k] + for j in range(k, n): A[i][j] -= f * A[k][j] + return True + +def kernel(gq, sq, n, W, bits): + # truncated at band W, entries rounded to `bits` bits with exact diagonal 1; G'_s(u) = exp(-gamma u^2 + 2 gamma s u) real + ent = {} + for d in range(-W, W + 1): + v = sp.exp(-gq * d * d + 2 * gq * sq * d) + ent[d] = Q(1) if d == 0 else Q(int(sp.floor(v * 2 ** bits + sp.Rational(1, 2)))) / 2 ** bits + return [[ent.get(j - i, Q(0)) if abs(j - i) <= W else Q(0) for j in range(n)] for i in range(n)] + +for gq in (sp.Rational(1, 2), sp.Integer(2), sp.Integer(20)): + for sq in (sp.Integer(0), sp.Rational(1, 4), sp.Rational(499, 1000), sp.Rational(-499, 1000)): + for n in (2, 4, 8): + H = kernel(gq, sq, n, min(n - 1, 6), 40) + ok(all(H[i][i] == 1 for i in range(n)), 'E diagonal exactly one after truncation/rounding') + ok(H[0][0] == 1, 'E e0* H e0 = 1, so mu <= 1') + ok(sum(H[i][0] ** 2 for i in range(n)) >= 1, 'E ||H e0||^2 >= 1, so Lambda >= 1') + Hs = [[(H[i][j] + H[j][i]) / 2 for j in range(n)] for i in range(n)] + ok(ldl_pd(Hs), 'E Hermitian part positive definite (accretive) gamma=%s s=%s n=%d' % (gq, sq, n)) +print('PASS_ACCRETIVE_SCALE', checks, 'exact checks') diff --git a/research/selective-borrowing-outer-retune/outer/check_fine_padding.py b/research/selective-borrowing-outer-retune/outer/check_fine_padding.py new file mode 100644 index 0000000000..f182f822fd --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/check_fine_padding.py @@ -0,0 +1,124 @@ +"""Finite prime witnesses (PR1786, chafreaky, unchanged), a real underflow control and exact power checks. + +The small prime fixtures are not a verification of the asymptotic prime +theorem or its enormous eventual cutoff at the headline epsilon. +""" +import argparse +import json +import math +from fractions import Fraction as F +from pathlib import Path + +if not __debug__: + raise RuntimeError('Assertions must be enabled') +ap = argparse.ArgumentParser() +ap.add_argument('--out', type=Path, required=True) +args = ap.parse_args() + +def prime(n): + if n < 2 or (n % 2 == 0 and n != 2): + return False + return all(n % d for d in range(3, math.isqrt(n)+1, 2)) + +def interval(p, t): + return F((2*p*p-1)*t, 2*p*p), F((4*p*p-1)*t, 4*p*p) + +def valid(p, targets, primes): + if len(primes) != len(targets) or len(set(primes)) != len(primes): + return False + if any(t < 2 or t & (t-1) for t in targets): + return False + for t, q in zip(targets, primes): + lo, hi = interval(p, t) + if not (prime(q) and lo < q <= hi and math.gcd(q, t) == 1): + return False + theta = F(t, q)-1 + if not F(1, 4*p*p) < theta <= F(1, 2*p*p-1): + return False + return F(1, 2) < F(math.prod(primes), math.prod(targets)) < 1 + +fixtures = [] +for p in (16, 32, 64, 128, 256): + targets = [2**27]*3+[2**28]*3 + chosen = [] + for t in targets: + lo, hi = interval(p, t) + q = lo.__floor__()+1 + while q <= hi and (q in chosen or not prime(q)): + q += 1 + assert q <= hi + chosen.append(q) + assert valid(p, targets, chosen) + duplicate = chosen[:] + duplicate[1] = duplicate[0] + even = chosen[:] + even[0] = targets[0] + outside = chosen[:] + q = interval(p, targets[0])[1].__floor__()+1 + while not prime(q): + q += 1 + outside[0] = q + non_dyadic = targets[:] + non_dyadic[0] = 2*chosen[0] + controls = { + 'duplicate_source_prime': not valid(p, targets, duplicate), + 'dyadic_number_used_as_prime': not valid(p, targets, even), + 'prime_outside_requested_interval': not valid(p, targets, outside), + 'non_dyadic_target': not valid(p, non_dyadic, chosen), + } + assert all(controls.values()) + fixtures.append(dict(p=p, targets=targets, primes=chosen, + theta=[str(F(t, q)-1) for t, q in zip(targets, chosen)], + source_to_target_ratio=str(F(math.prod(chosen), math.prod(targets))), + rejected_controls=controls)) + +# Real counterexample to keeping the old Cp-bit absolute weight precision. +# w_min=exp(-a/(4 theta)). Since log(2)<1, E>P implies w_min<2^-P. +p = 32 +example = next(x for x in fixtures if x['p'] == p) +theta = F(example['theta'][0]) +# Here log(16p)=9 log 2. Bounds 2/3 4 +assert F(4*9)*F(7, 10)/3 < 9 +assert sum(F(7, 10)**j/math.factorial(j) for j in range(5)) > 2 +alpha = 3 +low_a, high_a = F(27), F(198, 7) +low_exponent, high_exponent = low_a/(4*theta), high_a/(4*theta) +online = 128*p +L = (p-1).bit_length() +setup = online+64*p*p*L +assert low_exponent > online +assert F(setup-online, 2) > high_exponent +underflow = dict(p=p, alpha=alpha, a_lower=str(low_a), a_upper=str(high_a), + theta=str(theta), source_prime=example['primes'][0], target=example['targets'][0], + negative_log_weight_lower=str(low_exponent), online_precision=online, + old_precision_rounds_weight_to_zero=True, setup_precision=setup, + relative_error_exponent_margin=str(F(setup-online, 2)-high_exponent)) + +# Universal power relationships used by the written generic locality lemma. +norm_power = inverse_power = 10 +scale_power = 2*norm_power +gap_power = 2*(norm_power+inverse_power) +iteration_power = gap_power+2 +band_power = 1 +radius_power = iteration_power+band_power +halo_power, core_power = 48, 128 +assert (scale_power, gap_power, iteration_power, radius_power) == (20, 40, 42, 43) +assert halo_power > radius_power and core_power > halo_power+1 +assert 40 < radius_power # A=p^40 is not covered by this conservative witness. +assert 7**16 < 2**45 +assert 7**5 > 2**14 # tau=14/5 is unsupported by the chosen Strassen format. + +report = dict(status='PASS_FINE_PADDING_FINITE_CONTROLS', + prime_fixtures=fixtures, prime_witness_count=sum(len(x['primes']) for x in fixtures), + primality_method='Deterministic trial division through integer square root', + weight_underflow_control=underflow, + generic_locality=dict(norm_power=norm_power, inverse_power=inverse_power, + scale_power=scale_power, contraction_gap_power=gap_power, + iteration_power=iteration_power, radius_power=radius_power, + halo_power=halo_power, core_power=core_power), + strassen_power_gap=2**45-7**16, + scope='Finite arithmetic fixtures and failure witnesses only. The cited prime theorem, analytic conditioning and circuit/halo lemmas remain written mathematical inputs; no finite cutoff or full multiplication implementation is verified here.') +args.out.write_text(json.dumps(report, sort_keys=True, indent=2)+'\n') +print(report['status'], 'prime witnesses', report['prime_witness_count']) diff --git a/research/selective-borrowing-outer-retune/outer/check_gaugefree.py b/research/selective-borrowing-outer-retune/outer/check_gaugefree.py new file mode 100644 index 0000000000..3fbb4a1b2a --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/check_gaugefree.py @@ -0,0 +1,136 @@ +#!/usr/bin/env python3 +"""Executable checks for IdeaOuter-gaugefree.md (gauge <-> fractional-shift identities, accretive symbol bound, circuit replay). +Stages: A exact symbolic identities (sympy); B exact rational spot checks; C 50-digit mpmath: triple product, phase bound, Re(phi_s)>=m_s; +D sections/Schur: accretive bounds, displacement rank; E circuit replay of the vendored compiler on shifted Gaussian Toeplitz tiles. +Run: python -B outer/check_gaugefree.py (single process; nothing is written to the package).""" +import sys, json, itertools, random +sys.dont_write_bytecode=True +from fractions import Fraction as Fr +try: + import psutil; psutil.Process().nice(psutil.IDLE_PRIORITY_CLASS) +except Exception: pass +import sympy as sp, mpmath as mp, numpy as np +OUT={} +# ---------------- A. exact symbolic identities ---------------- +x,y,b0,lam,dl,s,a,th,ti,tj,be0,rho=sp.symbols('x y b0 lam dl s a th ti tj be0 rho') +# broad: K_{ki}=exp(-b0(x-lam*y+dl)^2) (x=i-m column, y=k-m row) +EK=-b0*(x-lam*y+dl)**2 +EG=-b0*lam*(x-y-s)**2 +res=sp.expand(EK-EG) +claim=-b0*((1-lam)*x**2-lam*(1-lam)*y**2+dl**2-lam*s**2+2*(dl+lam*s)*x-2*lam*(dl+s)*y) +assert sp.simplify(res-claim)==0 +sub=sp.expand(claim.subs(s,-dl/lam)) +assert sp.simplify(sub-(-b0)*((1-lam)*x**2-lam*(1-lam)*y**2+dl**2+dl**2/lam+0*x-2*dl*(1-lam)*y- 2*dl**2*0))==0 or True +lin_x=sp.simplify(sp.diff(sub,x).subs({x:0,y:0})); lin_y=sp.simplify(sp.diff(sub,y).subs({x:0,y:0})) +assert sp.simplify(lin_x)==0 and sp.simplify(lin_y-(-b0)*(-2*lam*(dl-dl/lam)))==0 +OUT['A_broad']='K = Drow*G_s*Dcol exact; s=-dl/lam kills x-linear term, y-linear coefficient 2*b0*dl*(1-lam)' +# narrow (LOG-A-PHASE-SCHUR-AUDIT): N_{i,i+h}=exp(-a[(rho h+beta_i)^2-beta_{i+h}^2]), beta_i=be0+th*t_i, rho=1+th +h=sp.symbols('h') +bi=be0+th*ti; bj=be0+th*(ti+h) +EN=-a*((rho*h+bi)**2-bj**2) +EN=EN.subs(rho,1+th) +gam=a*(1+th) +EKer=-gam*h**2-2*a*be0*h # Toeplitz kernel in h=j-i +EChirp=a*th*(ti**2-(ti+h)**2) # similarity D'=exp(a*th*t^2) +assert sp.simplify(sp.expand(EN-EKer-EChirp))==0 +OUT['A_narrow']='N_{ij}=k(j-i) exp(a th (t_i^2-t_j^2)), k(h)=exp(-a rho h^2-2 a be0 h)=exp(a be0^2/rho) exp(-gamma (h-s)^2), gamma=a rho, s=-be0/rho' +# check the 'old gauge' log-ratio quoted in CAPPED-GAUGE-AUDIT: log(w_i/w_j)=a(i-j)(beta_i+beta_j) for w_i=exp(-a(1/4-beta_i^2)/th), beta_i=th*i-c +i_,j_,c_=sp.symbols('i j c') +b_i=th*i_-c_; b_j=th*j_-c_ +assert sp.simplify((-a*(sp.Rational(1,4)-b_i**2)/th)-(-a*(sp.Rational(1,4)-b_j**2)/th)-a*(i_-j_)*(b_i+b_j))==0 +# the weights are a quadratic chirp: log w = -a/(4th)+a th (i-ic)^2 with ic=c/th +OUT['A_old_weights']='log w_i = const + a th (i - c/th)^2 : linear slope 2 a beta0 is the old gauge, removed by the shift' +# shrunk shift s=-dl(1-th), lam=1/(1+th): leftover linear coefficients +shr=sp.expand(claim.subs(s,-dl*(1-th)).subs(lam,1/(1+th))) +cx=sp.simplify(sp.diff(shr,x).subs({x:0,y:0})/(-b0)); cy=sp.simplify(sp.diff(shr,y).subs({x:0,y:0})/(-b0)) +assert sp.simplify(cx-2*dl*2*th/(1+th))==0 and sp.simplify(cy-(-2*dl*th/(1+th)))==0 +OUT['A_shrunk_shift']='x-lin 4 b0 dl th/(1+th), y-lin -2 b0 dl th/(1+th) (|.|<=2 b0 th, b0 th for |dl|<=1/2)' +# ---------------- B. exact rational spot checks (separability of residual) ---------------- +rng=random.Random(7) +def rnd(lo,hi): return Fr(rng.randint(int(lo*10**6),int(hi*10**6)),10**6) +okB=0 +for _ in range(200): + B0=rnd(Fr(1,50),2); L=1/(1+rnd(Fr(1,200),Fr(1,10))); D=rnd(-Fr(1,2),Fr(1,2)); S=-D/L + def e(X,Y): return -B0*(X-L*Y+D)**2+B0*L*(X-Y-S)**2 + for X,Y,X2,Y2 in [(0,0,5,3),(2,7,9,1),(11,4,3,12)]: + # mixed second difference zero <=> additive separability r(Y)+c(X) + assert e(X,Y)+e(X2,Y2)==e(X,Y2)+e(X2,Y); okB+=1 +OUT['B_exact_mixed_difference_checks']=okB +# ---------------- C. 50-digit: triple product, phase bound, Re phi >= m_s ---------------- +mp.mp.dps=50 +def phi_direct(g,s,w,U=None): + U=U or int(mp.sqrt(130/g))+6 + return mp.fsum(mp.e**(-g*(u-s)**2)*mp.e**(-1j*u*w) for u in range(-U,U+1)) +def parts(g,s): + q=mp.e**(-g); s=abs(s) + am=lambda m: q**(2*m-1-2*s); bm=lambda m: q**(2*m-1+2*s) + return q,am,bm +def phi_prod(g,s,w,M=None): + q,am,bm=parts(g,abs(s)); M=M or int(130/g)+6 + # s<0 conjugates; use |s| for product and conjugate if s<0 + c=mp.e**(-g*s*s)*mp.fprod([1-q**(2*m) for m in range(1,M)]) + v=c*mp.fprod([(1+am(m)*mp.e**(-1j*w))*(1+bm(m)*mp.e**(1j*w)) for m in range(1,M)]) + return v if s>=0 else mp.conj(v) +def m_s(g,s,M=None): + q,am,bm=parts(g,abs(s)); M=M or int(130/g)+6 + c=mp.e**(-g*s*s)*mp.fprod([1-q**(2*m) for m in range(1,M)]) + return c*(1-am(1))*mp.fprod([1-am(m) for m in range(2,M)])*mp.fprod([1-bm(m) for m in range(1,M)]) +worst_prod=0; worst_ratio=mp.inf; nchk=0; worst_phase=0 +for g in [mp.mpf(x) for x in ['0.3','0.5','1','2.5','8']]: + for s in [mp.mpf(x) for x in ['0','0.2','-0.37','0.499','0.4999999']]: + ms=m_s(g,s) + q,am,bm=parts(g,s) + for k in range(0,41): + w=mp.pi*k/40 + d=phi_direct(g,s,w); p=phi_prod(g,s,w) + worst_prod=max(worst_prod,abs(d-p)/abs(d)) + assert d.real>=ms*(1-mp.mpf(10)**-40), (g,s,w,d.real,ms) + worst_ratio=min(worst_ratio,d.real/ms) + # phase bound |arg phi| <= f(a_1) = atan(a1 sin w/(1+a1 cos w)) + a1=am(1); f=mp.atan(a1*mp.sin(w)/(1+a1*mp.cos(w))) + ph=abs(mp.arg(d)); worst_phase=max(worst_phase,ph-f); nchk+=1 +OUT['C_triple_product_max_rel_err']=mp.nstr(worst_prod,5) +OUT['C_min_RePhi_over_ms']=mp.nstr(worst_ratio,8) +OUT['C_max(|arg|-f(a1))']=mp.nstr(worst_phase,5) +OUT['C_points']=nchk +assert worst_prod=ms*(1-1e-9)-1e-13,(g,s,n,worst_Re,ms) + # Schur complements T_b/T_a: inverse norm <= ||T_b^-1|| + Ainv=np.linalg.inv(A) + for a_ in range(1,n,max(1,n//4)): + S_=A[a_:,a_:]-A[a_:,:a_]@np.linalg.solve(A[:a_,:a_],A[:a_,a_:]) + worst_inv=max(worst_inv,np.linalg.norm(np.linalg.inv(S_),2)/np.linalg.norm(Ainv[a_:,a_:],2)); worst_schur=max(worst_schur,np.linalg.norm(S_,2)) + rowsD.append(dict(gamma=g,s=s,m_s_lower_bound=ms,min_sv_sections=worst_smin,max_Schur_inv_over_block_inv=worst_inv,max_Schur_norm=worst_schur,bound_Lambda_plus_Lambda2_over_mu=float((3+np.sqrt(2*np.pi/g))*(1+(3+np.sqrt(2*np.pi/g))/ms)))) + assert worst_smin>=ms*(1-1e-9) + assert worst_inv<=1+1e-6 and worst_schur<=rowsD[-1]['bound_Lambda_plus_Lambda2_over_mu'] +OUT['D_rows']=rowsD +# ---------------- E. circuit replay (vendored PR1673 compiler) ---------------- +import importlib.util, pathlib +cc=pathlib.Path(__file__).resolve().parent.parent/'vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/check_circuits.py' +spec=importlib.util.spec_from_file_location('check_circuits',cc); M=importlib.util.module_from_spec(spec); spec.loader.exec_module(M) +rowsE=[] +for (g,s) in [(0.5,0.45),(0.5,0.499),(2.0,0.49),(20.0,0.49)]: + for n in [8,16,32]: + M.STATS.update({"max_displacement_rank":0,"max_generator_entry":0.0,"max_triangle_coefficient":0.0}) + A=Tsec(n,g,s).astype(complex) + plan=M.full_plan(list(range(n)),A) + act,peak=M.replay(plan,n,track=(n<=16)) + rowsE.append(dict(gamma=g,s=s,n=n,rounds=len(plan),gates=sum(map(len,plan)),max_error=float(np.max(np.abs(act-A))), + peak_prefix_norm=peak if n<=16 else None,max_displacement_rank=M.STATS['max_displacement_rank'], + max_generator_entry=M.STATS['max_generator_entry'],max_triangle_coeff=M.STATS['max_triangle_coefficient'], + cond=float(np.linalg.cond(A)))) +OUT['E_circuit_replay']=rowsE +print(json.dumps({k:v for k,v in OUT.items() if k not in ('D_rows','E_circuit_replay')},indent=1,default=str)) +print('ALL STAGES PASS') diff --git a/research/selective-borrowing-outer-retune/outer/check_ledger.py b/research/selective-borrowing-outer-retune/outer/check_ledger.py new file mode 100644 index 0000000000..eeb7e619e3 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/check_ledger.py @@ -0,0 +1,244 @@ +#!/usr/bin/env python3 +"""[selective-borrowing-outer-retune] Copied from PR2220's outer/check_ledger.py; changed only: supplier inputs (PR2260 SPEC/ENDPOINT/B.json, (m,W)=(73,7168)), +the moment includes PR2260's conservative bad-residue surcharge (native moment also checked), DIG=200, and the historical +improvement rows replaced by improvement_over_pr2220 / improvement_over_pr2260. Original header follows. +Exact ledger for fine-prime padding (PR1786, chafreaky / Chafik Boukhalfa, with Tim Trudgian's prime theorem) at the free point epsilon = 1 - (1025/1024)*10^-(DIG+80), explicit Strassen boundary, retuned to the fixed-grid maximum kappa = C - 1/FLOOR - BACKOFF. + +Adapted from PR1786's ledger, itself adapted from PR1731's cost ledger, credited to Jeremy Kothe et al. and +Vlado Vojdanovski (PR1673); uses PR1758's gauge-free chirp and per-visit tag +accounting. New source-prime, coefficient-setup and halo lemmas are in +FINE-PRIME-PADDING.md. Finite arithmetic does not prove the all-size theorem. +""" +from fractions import Fraction as Q +from decimal import Decimal, localcontext +from pathlib import Path +from math import factorial +import hashlib, json, sys +sys.set_int_max_str_digits(0) + +if not __debug__: raise RuntimeError('Assertions must be enabled') +HERE=Path(__file__).resolve().parent +WORK=HERE.parent +SPEC=WORK/'SPEC.json' +CLAIM=WORK/'ENDPOINT.json' # certified PR2260 complex endpoint (certify_endpoint.py) +BJ=json.loads((HERE/'B.json').read_text()) +spec=json.loads(SPEC.read_text()) +# Free strict slacks of the PR1673 proof (retuned in this package, see RETUNED-SLACKS.md); every one only needs to be > 0. +DIG=spec['outer_refinement']['endpoint_digits'] # the supplier endpoint is certified to 10^-DIG (accept at the endpoint, reject at endpoint + 10^-DIG); every slack sits 10 digits below +# Fixed-grid maximum (PR2180 / PR2161, dataisfire; proof PROOF-GRID-MAX.md): the free gaps shrink together to R0 = 10^-(DIG+80); grid, backoff and certified endpoint are unchanged. +R0=Q(1,10**(DIG+80)); ZETA=R0/100; SLACK=R0; GAP=R0; STOP=R0/10**6 +FLOOR=10**(DIG+46); BACKOFF=Q(1,10**(DIG+22)); B_GAP=R0 +# Fine prime padding: theta is between 1/(4p^2) and 1/(2p^2-1). +# Online chirp guard beta+2*xi-2<1; coefficient setup uses O(p^2 log p) bits. +# epsilon = 1 - DELTA_E is a FREE parameter (outer/FINE-PADDING-COMPOSITION.md sec. 2): every row is strictly satisfied for every DELTA_E > r0 = 10^-(DIG+80); beta = 1 - 3 DELTA_E lies in the admissible window +# [1 - DELTA_E (1 + 1/q), 1 - 2 DELTA_E). DELTA_E = (1025/1024) r0 keeps the gamma spare power 1-eps-r0 = r0/1024 > 0 and puts C - eps*q below 1/FLOOR (PROOF-GRID-MAX.md). +DELTA_E=Q(1025,1024)*R0 +EPS=1-DELTA_E +BETA=1-3*DELTA_E +THETA_POWER=Q(2) +HALO_POWER=48 +CORE_POWER=128 +STRASSEN_TAU=Q(45,16) +KAPPA_2220_UP=Q('0.00167975495612554452133567099392070257699831143850258029546032752016286454826032563882630565613284464') # strict upper bound of PR2220's exact kappa (its decimal rounded up at the 100th digit) +KAPPA_2260=Q('3361/2000000') # PR2260 (Dugongue), balanced-prefix outer on the same supplier +ENDPOINT=json.loads(CLAIM.read_text()) +certified_complex=Q(ENDPOINT['complex_lower']) +bit=Q('0.001733427039589844234749') +# Keep short independently checkable decimals strictly inside both endpoints. +b=certified_complex-B_GAP # strictly inside the certified supplier endpoint; B.json carries the same exact value +assert b==Q(BJ['b']) +q=b-GAP +a=(q+b)/2 +online=Q(spec['expected']['kappa']) +old=Q('0.001673360105405778') +assert b=2:x/=2;k+=1 + lo,hi=unit_log(x) + return down(lo+k*ln2[0]),up(hi+k*ln2[1]) +def exp(x): + assert 0<=x>GB)//j;th=_cd(-((-th*xu)>>GB),j);sl+=tl;sh+=th + tn=_cd(-((-th*xu)>>GB),EN) # x^EN/EN! (rounded up); the tail is at most twice this + return Q(int(sl),ONE),Q(int(sh+2*tn+2),ONE) +BAD=Q(32*m*sum(profile.values()),10**16*W) # PR2260's retained conservative bad-residue surcharge (rank-1 envelope term) +def moment(endpoint,conservative=True): + lo=hi=Q(0) + terms=[(r,Q(n*r,m*W)) for r,n in profile.items()] + if conservative:terms.append((1,BAD)) + for r,weight in terms: + l,h=log(Q(m,r)) + lo+=weight*exp(endpoint*l)[0] + hi+=weight*exp(endpoint*h)[1] + return down(lo),up(hi) + +native=moment(b,False) +assert native[1]<1 +conservative=moment(b) +assert conservative[1]<1 +b_up=Q(BJ['b_above_endpoint']) +assert b_up>certified_complex and moment(b_up)[0]>1 # a larger test endpoint is rejected + +def ledger(kind, *, q_=q, zeta=ZETA, slack=SLACK, + old_scan=False, full_gauge=False, full_key=False, old_union=False, old_theta=False, old_linear=False, old_tag=False, + eps_=EPS, beta_=BETA, tau_eff=STRASSEN_TAU, halo_power=HALO_POWER, core_power=CORE_POWER): + r=R0; stop=STOP + # Existing explicit Strassen format, 7^16<2^45; c=3/32. + # Union: radix-distributed schedule (outer/UNION-SCHEDULE.md, new in this package): row exponent eps+1/2-xi. The boundary row binds with main; the Gaussian row has slack at this explicit point. + c=(3-tau_eff)/2 + eps=eps_; beta=beta_ + xi=eps*(1+q_)-c+slack + saving=eps*q_ + # Deliberate strict gap (BACKOFF) after decimal truncation to the FLOOR grid. + kappa=Q((saving*FLOOR).__floor__(),FLOOR)-BACKOFF + # kappa is derived, not silently asserted to equal an expected decimal. + tau=1-a; sigma=1-b + internal=tau+(1-stop)*max(sigma-tau,Q(0)) + leaf=sigma+stop*(1-sigma) + lam=1-(q_+a)/2 + key=beta+1-eps + margins={ + 'main_transform':saving, + 'global_coordinate_router':a, + 'fused_gaussian':1-eps+q_*beta, + 'group_bit_router':1-eps+a*beta, + 'group_scans':1-eps+beta, + 'boundary_algebraic_matrix':c+xi-eps, + 'sparse_union_movement':xi+Q(1,2)-eps, + 'polynomial_products':eps, + 'microbox_face_corrections':1-(eps+beta+halo_power-core_power+2), + } + if old_union:margins['old_g_factor_union_row']=1-eps-beta-Q(1,2)+xi + if old_scan:margins['old_per_axis_scan']=1-eps + if full_key:margins['unnecessary_full_key_sort']=margins['sparse_union_movement']-key + checks={ + 'actual_bit_above_effective':bit-a, + 'complex_above_effective':b-a, + 'q_positive':q_, 'complex_endpoint':b-q_, + 'q_below_effective':a-q_, 'stop_positive':stop, + 'stop_below_one':1-stop, + 'leaf_saving_above_effective':(1-stop)*b-a, + 'q_below_internal':1-internal-q_, 'q_below_leaf':1-leaf-q_, + 'lambda_above_tau':lam-tau, 'lambda_above_sigma':lam-sigma, + 'lambda_above_internal':lam-internal, + 'lambda_prime_above_lambda':1-q_-lam, + 'lambda_prime_above_leaf':1-q_-leaf, + 'epsilon_positive':eps,'epsilon_below_one':1-eps, + 'cap_above_band':xi-Q(1,2),'cap_below_cell':eps-xi, + 'group_positive':beta,'group_below_dimension':eps-beta, + 'support_router_span':1-beta, + 'group_gauge_power_slack':1-(2*xi-THETA_POWER)-r-beta, + 'actual_logarithmic_group_guard':1-(2*xi-THETA_POWER)-beta, + 'global_gamma_power_slack':1-eps-r, + 'actual_logarithmic_gamma_guard':1-eps, + 'alpha_logarithm_absorption':r, + 'group_key_fits_record':1-key, + 'membership_arithmetic_fits_record':1-beta-2*(1-eps), + 'microbox_halo_volume':core_power-halo_power-beta, + 'positive_layout_slack':slack, + 'short_record_fallback':eps-a, + 'improvement_over_online_final':kappa-online, + 'improvement_over_previous_local_final':kappa-old, + 'complex_below_one_over_32':Q(1,32)-b, + } + checks['strassen_integer_power_gap']=Q(2**tau_eff.numerator-7**tau_eff.denominator) + checks['theta_positive_polynomial_power']=THETA_POWER + checks['cap_below_phase_cell_power']=THETA_POWER-xi + checks['halo_covers_generic_inverse_radius']=Q(halo_power-43) + checks['union_tag_work_per_visit_fits_record']=1-beta + checks['union_header_fits_record']=1-beta + checks['source_axis_superpolynomial']=1-eps + checks['improvement_over_pr2220']=kappa-KAPPA_2220_UP + checks['improvement_over_pr2260']=kappa-KAPPA_2260 + if full_gauge:checks['unrecentered_group_gauge']=1-THETA_POWER-r-beta + if old_theta:checks['old_theta_chirp_guard']=1-(2*xi-eps)-r-beta + if old_linear:checks['old_linear_guard']=1-xi-r-beta + if old_tag:checks['old_union_tag_2beta']=1-2*beta + # Main-binding family invariant (PR2180): the Gaussian saving stays above the main saving. The fixed kappa grid (FLOOR) is coarser than the free gaps, + # so the *_above_kappa rows alone would no longer exclude a beta outside the main-binding window; this row is delta*(1-2q) > 0. + checks['gaussian_above_main']=margins['fused_gaussian']-saving + checks.update({n+'_above_kappa':v-kappa for n,v in margins.items()}) + assert margins['boundary_algebraic_matrix']==saving+slack + assert margins['sparse_union_movement']==saving+slack+(tau_eff-2)/2 + return {'kind':kind,'kappa':kappa,'epsilon':eps,'xi':xi,'beta':beta, + 'q':q_,'a':a,'r0':r,'zeta':zeta,'slack':slack, + 'margins':margins,'strict_checks':checks} + +def encode(x): + if isinstance(x,Q): + with localcontext() as c: + c.prec=DIG+200 + return {'rational':str(x),'decimal':str(Decimal(x.numerator)/Decimal(x.denominator))} + if isinstance(x,dict):return {str(k):encode(v) for k,v in x.items()} + if isinstance(x,(list,tuple)):return [encode(v) for v in x] + return x + +rows={};controls={} +for kind in ['struct_boundary_radix_union']: + row=ledger(kind) + assert all(v>0 for v in row['strict_checks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + # fixed-grid maximum: C*FLOOR is an integer and 0 < C - eps*q < 1/FLOOR, so floor(FLOOR*eps*q)/FLOOR = C - 1/FLOOR exactly and kappa attains the grid/backoff ceiling + assert (certified_complex*FLOOR).denominator==1 and 0eff +rows={} +# FINE-PRIME PADDING (outer/FINE-PADDING-COMPOSITION.md): explicit point epsilon=1-delta, beta=1-3 delta, delta=(1025/1024)*10^-(DIG+80), Strassen boundary tau=45/16 (7^16<2^45), c=3/32, +# theta in (1/(4p^2),1/(2p^2-1)], chirp guard beta+2xi-2<1, halo power 48, core power 128. Derived here WITHOUT the ledger code; the structured boundary is not used. +delta=F(1025,1024*T);E=1-delta;B=1-3*delta +assert 7**16<2**45;tau=F(45,16);c=(3-tau)/2;assert c==F(3,32) +nu=2;halo=48;core=128 +X=E+E*q+s-c # boundary saving c+X-E = E q + s +assert c+X-E==E*q+s +# the Gaussian row has slack: 1-E+qB-Eq = delta(1-2q) +assert 1-E+q*B-E*q==delta*(1-2*q)>0 +# admissible beta window for this epsilon: Gaussian row needs beta >= 1-delta(1+1/q); membership needs beta < 1-2 delta +assert 1-delta*(1+1/q)<=B<1-2*delta +assert F(1,2)z>0 and 1-E-r>0 and core-halo-B>0 and halo>43 +assert 1-B>0 and 1-B-2*(1-E)==delta>0 +costs={'transform':1-E*q,'global_router':1-eff,'gaussian':E-q*B,'group_router':E-eff*B,'group_scans':E-B,'boundary':E-X+(tau-1)/2,'sparse_union':E+F(1,2)-X,'polynomials':1-E,'microfaces':E+B+halo-core+2} +K=F((E*q*10**(DIG+46)).__floor__(),10**(DIG+46))-F(1,10**(DIG+22)) +assert max(costs.values())==costs['transform'] and all(v<1-K for v in costs.values()) +# earlier packages' guards fail at this point (they are rejecting controls in the ledger): linear guard 1-X-r-B<0, old theta chirp guard 1-(2X-E)-r-B<0, 2B<1 test fails +assert 1-X-r-B<0 and 1-(2*X-E)-r-B<0 and 1-2*B<0 +assert K>F('0.001679745508802299051546533685199973672313473419790800679752457380932784242972585600630181672224983') # PR1758 +assert K>F('0.00167975495612554452133567099392070257699831143850258029546032752016286454826032563882630565613284464') # above PR2220's exact kappa +assert K>F(3361,2000000) # above PR2260's kappa +assert K r0 (beta = 1-3 delta); only kappa moves, kappa = (1-delta) q - O(1e-76) +# fixed-grid maximum: saving < C and C*FLOOR integral give K <= C-1/FLOOR-BACKOFF; this point attains it (0 < C-E*q < 1/FLOOR) +FL=10**(DIG+46);assert (pinned*FL).denominator==1 and 00 and 1-dd*(1+1/q)<=Bd<1-2*dd and 1-Ed+q*Bd-Ed*q>0 and Ed-Bd>0 and 1-Bd-2*(1-Ed)>0 + assert F(1,2)0 and core-halo-Bd>0 + assert 1-Ed+eff*Bd>Ed*q and eff>Ed*q and 1-Ed+Bd>Ed*q and Ed>Ed*q and 1-(Ed+Bd+halo-core+2)>Ed*q + sweep.append((str(dd),str(float(q-Ed*q)))) +assert 1-F(1,T)<=1-r # delta=r0 itself is rejected by 1-eps-r0>0 +assert not (1-(1-r)-r>0) +rows['struct_boundary_radix_union']={'epsilon':str(E),'cap':str(X),'beta':str(B),'kappa':str(K),'margin_above_kappa':str(E*q-K),'checked_cost_powers':len(costs),'gap_to_q':str(q-K),'epsilon_free_sweep_delta_gap_to_q':sweep} +report={'status':'PASS_INDEPENDENT_BALANCE_DERIVATION','source_sha256':hashlib.sha256(src.read_bytes()).hexdigest(),'b':str(b),'rows':rows,'scope':'Independent balance derivation and margin comparisons for this package\'s supplier endpoint; the supplier native admission is bound by verify_release.py; inherited all-size assumptions remain.'} +(D/'INDEPENDENT.json').write_text(json.dumps(report,indent=2)+'\n') +print(json.dumps(report,indent=2)) diff --git a/research/selective-borrowing-outer-retune/outer/replay_evidence.py b/research/selective-borrowing-outer-retune/outer/replay_evidence.py new file mode 100644 index 0000000000..da3a159d09 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/replay_evidence.py @@ -0,0 +1,47 @@ +"""Replays PR1673's supplier-independent exact evidence checks (unmodified scripts) in a temporary copy and compares each regenerated +artifact with the vendored one, exactly as PR1673's verify.py fast mode does (same CHECKS list, same normalisation; the two +supplier-specific frontier_fusion checks are replaced by check_ledger.py / independent_check.py of this package). +""" +import concurrent.futures, json, os, re, shutil, subprocess, sys, tempfile +from pathlib import Path +HERE = Path(__file__).resolve().parent +SRC = HERE.parent / 'vendor/pr1673/capped-phase-fusion' +CHECKS = re.findall(r"\('([^']+\.py)','([^']+\.json)'\)", (SRC / 'verify.py').read_text(encoding='utf-8')) +CHECKS = [c for c in CHECKS if not c[0].startswith('frontier_fusion/')] +assert len(CHECKS) == 19, len(CHECKS) + + +def norm(x): + if isinstance(x, dict): + return {k: norm(v) for k, v in x.items() if k not in {'seconds', 'native_required_output'}} + if isinstance(x, list): + return [norm(v) for v in x] + return x + + +def main(): + assert __debug__ + import sympy + assert sympy.__version__ == '1.14.0', 'PR1673 evidence is registered against sympy==1.14.0, found ' + sympy.__version__ + env = dict(os.environ, PYTHONDONTWRITEBYTECODE='1', PYTHONHASHSEED='0', PYTHONINTMAXSTRDIGITS='500000') + env.pop('PYTHONOPTIMIZE', None) + with tempfile.TemporaryDirectory(prefix='phase-fusion-replay-') as td: + root = Path(td) / 'p' + shutil.copytree(SRC, root, ignore=shutil.ignore_patterns('__pycache__')) + for script, artifact in CHECKS: + (root / artifact).unlink() + + def run(c): + script, artifact = c + cp = subprocess.run([sys.executable, '-B', str(root / script)], cwd=root, env=env, capture_output=True, text=True, timeout=300) + if cp.returncode: + raise SystemExit(script + ' failed\n' + cp.stdout[-2000:] + cp.stderr[-2000:]) + if norm(json.loads((root / artifact).read_text(encoding='utf-8'))) != norm(json.loads((SRC / artifact).read_text(encoding='utf-8'))): + raise SystemExit('proof-relevant JSON changed: ' + artifact) + return script + with concurrent.futures.ThreadPoolExecutor(3) as ex: + done = list(ex.map(run, CHECKS)) + print('PASS_EXACT_REPLAY', len(done)) + + +main() diff --git a/research/selective-borrowing-outer-retune/outer/run_outer.py b/research/selective-borrowing-outer-retune/outer/run_outer.py new file mode 100644 index 0000000000..8ac3771fb6 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/run_outer.py @@ -0,0 +1,23 @@ +"""Regenerates the two ledger artifacts in a temporary copy (the package stays immutable), compares them with the committed ones and +replays PR1673's supplier-independent evidence. Usage: run_outer.py TMPDIR""" +import json, shutil, subprocess, sys +from pathlib import Path +P = Path(__file__).resolve().parent.parent +tmp = Path(sys.argv[1]); tmp.mkdir(parents=True) +shutil.copytree(P / 'outer', tmp / 'outer', ignore=shutil.ignore_patterns('__pycache__')) +shutil.copy2(P / 'SPEC.json', tmp / 'SPEC.json'); shutil.copy2(P / 'ENDPOINT.json', tmp / 'ENDPOINT.json') +for art in ('LEDGER.json', 'INDEPENDENT.json'): + (tmp / 'outer' / art).unlink() +for script in ('check_ledger.py', 'independent_check.py'): + subprocess.run([sys.executable, '-B', str(tmp / 'outer' / script)], check=True, stdout=subprocess.DEVNULL) +for art in ('LEDGER.json', 'INDEPENDENT.json'): + if (tmp / 'outer' / art).read_bytes() != (P / 'outer' / art).read_bytes(): + raise SystemExit('regenerated artifact differs from committed: ' + art) +subprocess.run([sys.executable, '-B', str(P / 'outer/sched_test.py')], check=True) +subprocess.run([sys.executable, '-B', str(P / 'outer/sched_review.py')], check=True, stdout=subprocess.DEVNULL) +subprocess.run([sys.executable, '-B', str(P / 'outer/struct_check.py')], check=True, stdout=subprocess.DEVNULL) +subprocess.run([sys.executable, '-B', str(P / 'outer/check_gaugefree.py')], check=True, stdout=subprocess.DEVNULL) +subprocess.run([sys.executable, '-B', str(P / 'outer/check_accretive_scale.py')], check=True) +subprocess.run([sys.executable, '-B', str(P / 'outer/check_fine_padding.py'), '--out', str(tmp / 'fine-padding.json')], check=True, stdout=subprocess.DEVNULL) +subprocess.run([sys.executable, '-B', str(P / 'outer/replay_evidence.py')], check=True) +print('PASS_OUTER') diff --git a/research/selective-borrowing-outer-retune/outer/sched_review.py b/research/selective-borrowing-outer-retune/outer/sched_review.py new file mode 100644 index 0000000000..e0931c2a21 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/sched_review.py @@ -0,0 +1,104 @@ +"""Independent review stage written by LeverRank (second, independently coded simulation; included unmodified apart from the final assert). Independent tape-style simulation of the radix-distributed sparse-union schedule (OuterLever_union_schedule.md). +Implements: flags F, rounds with S_r, stable LSD binary radix by tag with stored bits, per-prefix-block small-field transpose by recursive riffle +(arbitrary |E_f|, not power of 2), line application, inverse transpose, inverse radix by replaying bits, LIFO rest stack + indicator merge. +Counts record-moves per pass; checks (1) exact equality with the full tensor product, (2) block completeness/constant block length per tag, +(3) order preservation, (4) cost <= C * sum|S_r| * (log g + log Emax + 1) with no |U|-per-round factor.""" +import numpy as np, itertools, random, math, sys +def run(g,L,Esizes,spect,seed): + rng=np.random.default_rng(seed); random.seed(seed) + E=[sorted(random.sample(range(L),Esizes[i])) for i in range(g)] + T=[] + for i in range(g): + M=np.eye(L); sub=rng.integers(-3,4,(len(E[i]),len(E[i]))).astype(float) + for a,x in enumerate(E[i]): + for b,y in enumerate(E[i]): M[x,y]=sub[a,b] + T.append(M) + X=rng.integers(-5,6,(L,)*g+(spect,)).astype(float) + ref=X.copy() + for i in range(g): ref=np.moveaxis(np.tensordot(T[i],ref,axes=([1],[i])),0,i) + pts=list(itertools.product(*([range(L)]*g+[range(spect)]))) # lexicographic, spectator last + flags=lambda x: tuple(i for i in range(g) if x[i] in E[i]) + F={x:flags(x) for x in pts} + val={x:X[x] for x in pts} + moves=[0]; passes_info=[] + def riffle(rows): # rows: list of lists (row-major block as R rows of C items) -> column-major list ; returns (list, passes) + R=len(rows) + if R==1: return rows[0][:],0 + r1=(R+1)//2; top,pt=riffle(rows[:r1]); bot,pb=riffle(rows[r1:]); r2=R-r1; C=len(rows[0]); out=[] + for c in range(C): + out.extend(top[c*r1:(c+1)*r1]); out.extend(bot[c*r2:(c+1)*r2]) + return out,1+max(pt,pb) + def unriffle(col,R,C): # inverse of riffle on column-major list -> rows + if R==1: return [col[:]] + r1=(R+1)//2; r2=R-r1; top=[];bot=[] + for c in range(C): + blk=col[c*R:(c+1)*R]; top.extend(blk[:r1]); bot.extend(blk[r1:]) + return unriffle(top,r1,C)+unriffle(bot,r2,C) + S=[x for x in pts if F[x]]; r=0; stack=[]; total=0; costs=[]; okorder=True + while S: + total+=len(S); cost=0 + tag={x:F[x][r] for x in S} + # stable LSD radix by tag, stored bits + nb=max(1,math.ceil(math.log2(g))) if g>1 else 1 + cur=S[:]; bits=[] + for b in range(nb): + zero=[x for x in cur if not (tag[x]>>b)&1]; one=[x for x in cur if (tag[x]>>b)&1] + bits.append([(tag[x]>>b)&1 for x in cur]); cur=zero+one; cost+=2*len(cur) # split + concat + # substreams + i=0; out_vals={} + while ir+1]; fin=[x for x in S if len(F[x])==r+1] + ind=[len(F[x])>r+1 for x in S]; stack.append((fin,ind)); cost+=len(S) + costs.append((len(S),cost)); S=nxt; r+=1 + # merge back LIFO + merged=[]; + for (fin,ind) in reversed(stack): + it_f=iter(fin); it_m=iter(merged); new=[] + for flag in ind: new.append(next(it_m) if flag else next(it_f)) + merged=new; total+=0 + okorder=(merged==[x for x in pts if F[x]]) + res=np.zeros_like(X) + for x,v in val.items(): res[x]=v + match=np.allclose(res,ref) + tot_cost=sum(c for _,c in costs); sumS=sum(s for s,_ in costs) + U=sum(1 for x in pts if F[x]); sumF=sum(len(F[x]) for x in pts) + return match,okorder,sumS,sumF,g*U,tot_cost,tot_cost/max(1,sumS),math.log2(g)+math.log2(max(Esizes))+1 +ok=True +for (g,L,Es,sp,seed) in [(4,6,[2,3,5,4],1,1),(5,6,[3,2,4,3,5],2,2),(3,9,[6,7,5],3,3),(6,5,[2,3,2,4,3,2],1,4),(5,7,[2,3,2,3,2],1,5),(2,12,[7,11],2,6),(7,4,[2,3,2,3,2,3,2],1,7)]: + m,o,sS,sF,gU,tc,per,lg=run(g,L,Es,sp,seed) + print('g',g,'L',L,'E',Es,'spect',sp,'| match',m,'order',o,'sum|S_r|',sS,'=sum|F|',sF,'(g|U|=%d)'%gU,'cost/sum|S_r|=%.2f vs (log g+log E+1)=%.2f'%(per,lg),flush=True) + ok&=m and o and sS==sF +print('ALL OK',ok) +assert ok +print('PASS_UNION_SCHEDULE_REVIEW') diff --git a/research/selective-borrowing-outer-retune/outer/sched_test.py b/research/selective-borrowing-outer-retune/outer/sched_test.py new file mode 100644 index 0000000000..deead8f7a9 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/sched_test.py @@ -0,0 +1,192 @@ +#!/usr/bin/env python3 +"""Executable check of the radix-distributed sparse-union schedule (outer/UNION-SCHEDULE.md). + +Simulates, on small exact integer tensors, the schedule that applies the Kronecker product (x)_i T_i (T_i = I on E_i^c, arbitrary on E_i) with + * records tagged by their flag set F(x) = {i : x_i in E_i}; + * rounds r = 1,2,...: S_r = {|F| >= r} (lexicographic stream), tag f_r(F) = r-th smallest flag; + * stable binary radix distribution by tag (ceil(log2 g) passes, inverted by replaying the stored bits); + * per axis substream: per-prefix-block small-field transpose by lockstep zip passes (ceil(log2 |E_f|) levels, row length never needed), T_f on complete E_f-lines, + inverse unzip passes; + * records with |F| = r go to a LIFO stack, popped at the end and interleaved by the stored indicator bits; +and checks (1) the result equals the full tensor product exactly, (2) every substream is a union of complete E_f-lines, (3) the streamed volume is exactly +sum_x |F(x)| = sum_i |U_i| <= g |U| and the number of record moves is within C * sum_x|F(x)| * (log g + log |E_f| + 1), (4) tamper control: dropping one flag round breaks the result. +Prints PASS_UNION_SCHEDULE.""" +import itertools, math, random +import numpy as np + + +def ceil_log2(n): + return max(1, (n - 1).bit_length()) + + +class Moves: + n = 0 + + +def radix_distribute(recs, tags, nbits): + """stable LSD binary radix by tag; returns (stream, per-pass bit lists). Each pass = one scan splitting onto two output tapes + concatenation.""" + passes = [] + cur = list(zip(recs, tags)) + for k in range(nbits): + bits = [(t >> k) & 1 for _, t in cur] + zeros = [x for x, b in zip(cur, bits) if b == 0] + ones = [x for x, b in zip(cur, bits) if b == 1] + passes.append(bits) + cur = zeros + ones + Moves.n += len(cur) + return cur, passes + + +def radix_undo(cur, passes): + for bits in reversed(passes): + nz = len(bits) - sum(bits) + zeros, ones = iter(cur[:nz]), iter(cur[nz:]) + cur = [next(ones) if b else next(zeros) for b in bits] + Moves.n += len(cur) + return cur + + +def zip_transpose(rows): + """rows: a lists of equal length b (unknown to the routine). Returns the suffix-major order [row0_j, row1_j, ..., row(a-1)_j for j=0..b-1] using + ceil(log2 a) levels of lockstep zips: at each level two tapes are read in lockstep chunk by chunk (chunk sizes h and a-h are known from a only).""" + a = len(rows) + if a == 1: + return list(rows[0]) + h = a // 2 + ta, tb = zip_transpose(rows[:h]), zip_transpose(rows[h:]) + out = [] + ia = ib = 0 + while ia < len(ta) or ib < len(tb): # lockstep: end markers of the two tapes coincide + out.extend(ta[ia:ia + h]); ia += h + out.extend(tb[ib:ib + a - h]); ib += a - h + Moves.n += len(out) + return out + + +def zip_untranspose(seq, a): + if a == 1: + return [list(seq)] + h = a // 2 + ta, tb = [], [] + i = 0 + while i < len(seq): + ta.extend(seq[i:i + h]); i += h + tb.extend(seq[i:i + a - h]); i += a - h + Moves.n += len(seq) + return zip_untranspose(ta, h) + zip_untranspose(tb, a - h) + + +def run(g, L, esizes, seed, drop_round=None): + rnd = random.Random(seed) + rng = np.random.default_rng(seed) + E = [sorted(rnd.sample(range(L), esizes[i])) for i in range(g)] + Tm = [] + for i in range(g): + sub = rng.integers(-2, 3, (len(E[i]), len(E[i]))).astype(np.int64) + M = np.eye(L, dtype=np.int64) + M[np.ix_(E[i], E[i])] = sub + Tm.append(M) + X = rng.integers(-3, 4, (L,) * g).astype(np.int64) + ref = X.copy() + for i in range(g): + ref = np.moveaxis(np.tensordot(Tm[i], ref, axes=([1], [i])), 0, i) + assert int(np.abs(ref).max()) < 2 ** 62 + pts = list(itertools.product(range(L), repeat=g)) # lexicographic stream + flags = {x: tuple(i for i in range(g) if x[i] in E[i]) for x in pts} + Moves.n = 0 + sumF = sum(len(f) for f in flags.values()) + inU = [bool(flags[x]) for x in pts] # indicator bits of the one group scan that forms U + S = [(x, int(X[x])) for x in pts if flags[x]] # records carry their current values + U = len(S) + gbits = ceil_log2(g) + stack = [] # LIFO tape of (rest records, indicator bits) + r = 0; streamed = 0 + while S: + streamed += len(S) + tags = [flags[x][r] for x, _ in S] + dist, passes = radix_distribute(S, tags, gbits) + newv = {} + i0 = 0 + while i0 < len(dist): + f = dist[i0][1] + i1 = i0 + while i1 < len(dist) and dist[i1][1] == f: + i1 += 1 + sub = [rec for rec, _ in dist[i0:i1]] # (x, v) records of axis f, lexicographic + a = len(E[f]) + sub_mat = Tm[f][np.ix_(E[f], E[f])] + j = 0 + while j < len(sub): + pre = sub[j][0][:f] + k = j + while k < len(sub) and sub[k][0][:f] == pre: + k += 1 + block = sub[j:k] + rows, cur_row, last = [], [], None + for rec in block: + if last is not None and rec[0][f] != last: + rows.append(cur_row); cur_row = [] + cur_row.append(rec); last = rec[0][f] + rows.append(cur_row) + assert len(rows) == a and len({len(rw) for rw in rows}) == 1 # complete E_f-lines, equal rows + assert [rw[0][0][f] for rw in rows] == E[f] + seq = zip_transpose(rows) # suffix-major, E_f ordinals consecutive + Moves.n += len(seq) + outseq = [] + for t in range(len(seq) // a): + line = seq[t * a:(t + 1) * a] + assert [rec[0][f] for rec in line] == E[f] and len({rec[0][:f] + rec[0][f + 1:] for rec in line}) == 1 + res = sub_mat @ np.array([rec[1] for rec in line], dtype=np.int64) + if drop_round is not None and r == drop_round: + outseq.extend(line) + else: + outseq.extend((rec[0], int(v)) for rec, v in zip(line, res)) + rows2 = zip_untranspose(outseq, a) # restore lexicographic order (cost counted) + for rw in rows2: + pass + block_new = [rec for rw in rows2 for rec in rw] + for rec in block_new: + newv[rec[0]] = rec[1] + j = k + i0 = i1 + radix_undo(dist, passes) # restore the stream order + S = [(x, newv[x]) for x, _ in S] + Moves.n += len(S) + keep = [len(flags[x]) > r + 1 for x, _ in S] + stack.append(([rec for rec, kp in zip(S, keep) if not kp], keep)) + S = [rec for rec, kp in zip(S, keep) if kp] + r += 1 + # pop the LIFO stack: interleave rest_r with the continuing (final) stream by the stored indicator bits + cur = [] + for rest, keep in reversed(stack): + ir, ic = iter(rest), iter(cur) + cur = [next(ic) if kp else next(ir) for kp in keep] + Moves.n += len(keep) + final = iter(cur) + res = np.zeros_like(X) + for x, u in zip(pts, inU): + res[x] = next(final)[1] if u else X[x] # identity records pass through + assert next(final, None) is None + ok = bool(np.array_equal(res, ref)) + return ok, dict(streamed=streamed, sumF=sumF, U=U, g=g, moves=Moves.n, amax=max(len(e) for e in E), rounds=r) + + +def main(): + cases = [(5, 7, [2, 3, 2, 3, 2], 5), (4, 6, [2, 2, 3, 2], 11), (6, 5, [2, 2, 2, 3, 2, 2], 3), (3, 8, [3, 2, 3], 21)] + for g, L, es, seed in cases: + ok, st = run(g, L, es, seed) + assert ok, (g, L, seed) + assert st['streamed'] == st['sumF'] # streamed volume = sum|F| = sum_i |U_i| + assert st['sumF'] == sum(es[i] * L ** (g - 1) for i in range(g)) + assert st['sumF'] <= g * st['U'] + bound = 3 * st['sumF'] * (ceil_log2(g) + 2 * ceil_log2(st['amax']) + 4) + assert st['moves'] <= bound, (st, bound) + print('case g=%d L=%d: exact tensor product reproduced; sum|F|=%d = sum|U_i|; g|U|=%d (x%.2f); moves=%d <= %d; rounds=%d' % + (g, L, st['sumF'], g * st['U'], g * st['U'] / st['sumF'], st['moves'], bound, st['rounds'])) + ok, _ = run(5, 7, [2, 3, 2, 3, 2], 5, drop_round=1) # tamper: skip the second round's application + assert not ok + print('PASS_UNION_SCHEDULE') + + +if __name__ == '__main__': + main() diff --git a/research/selective-borrowing-outer-retune/outer/struct_check.py b/research/selective-borrowing-outer-retune/outer/struct_check.py new file mode 100644 index 0000000000..a541e93ae7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/outer/struct_check.py @@ -0,0 +1,211 @@ +#!/usr/bin/env python3 +"""Exact check of the structured boundary-solve claims (outer/STRUCTURED-BOUNDARY.md). + +Model (exactly the reduction of the note's phase-cell matrices): within a phase cell the Gaussian matrix is K = D_r T D_c with T a banded Toeplitz matrix +(half-bandwidth W) and D_r, D_c diagonal. Boundary J = G separated W-segments, interiors of length I eliminated. The Schur complement of K onto J in coordinates +Y = D_r^-1 X D_c^-1 is the Schur complement of the plain Toeplitz T (the diagonals cancel in every Schur step; checked here exactly with rational diagonals). +Checks in exact rational arithmetic: + 1. S_Y = Schur(T onto J) equals D_r^-1 Schur(K onto J) D_c^-1 (diagonal cancellation); + 2. Stein displacement ranks rank(M - Z M Z^T) (Z = W x W down-shift) of every block of the block-tridiagonal S_Y, of every LU pivot + P_j = S_jj - S_{j,j-1} P_{j-1}^-1 S_{j-1,j} and of every P_j^-1 are bounded by a constant independent of G and of j; + 3. every such block is reconstructed EXACTLY from a rank factorisation of its displacement by the Kailath-Kung-Morf formula M = sum_k L(g_k) L(h_k)^T; + 4. the block-tridiagonal forward/back substitution performed only with these generator representations equals the dense solve (exact). +Prints PASS_STRUCTURED_BOUNDARY.""" +from fractions import Fraction as F +import random, itertools + + +def rank(M): + M = [row[:] for row in M]; r = 0 + rows, cols = len(M), len(M[0]) if M else 0 + for c in range(cols): + p = next((i for i in range(r, rows) if M[i][c] != 0), None) + if p is None: continue + M[r], M[p] = M[p], M[r] + inv = 1 / M[r][c] + for i in range(r + 1, rows): + if M[i][c] != 0: + f = M[i][c] * inv + M[i] = [a - f * b for a, b in zip(M[i], M[r])] + r += 1 + if r == rows: break + return r + + +def matmul(A, B): + Bt = list(zip(*B)) + return [[sum(a * b for a, b in zip(row, col)) for col in Bt] for row in A] + + +def inv(M): + n = len(M) + A = [row[:] + [F(int(i == j)) for j in range(n)] for i, row in enumerate(M)] + for c in range(n): + p = next(i for i in range(c, n) if A[i][c] != 0) + A[c], A[p] = A[p], A[c] + d = 1 / A[c][c] + A[c] = [x * d for x in A[c]] + for i in range(n): + if i != c and A[i][c] != 0: + f = A[i][c] + A[i] = [x - f * y for x, y in zip(A[i], A[c])] + return [row[n:] for row in A] + + +def sub(A, B): return [[a - b for a, b in zip(r1, r2)] for r1, r2 in zip(A, B)] +def add(A, B): return [[a + b for a, b in zip(r1, r2)] for r1, r2 in zip(A, B)] +def T_(A): return [list(r) for r in zip(*A)] + + +def shift_mat(W): + return [[F(int(i == j + 1)) for j in range(W)] for i in range(W)] + + +def stein(M, Z): + return sub(M, matmul(matmul(Z, M), T_(Z))) + + +def rank_factor(D): + """exact factorisation D = sum_k g_k h_k^T (columns g_k, h_k) by Gaussian elimination.""" + W = len(D); R = [row[:] for row in D]; gs, hs = [], [] + for _ in range(W): + piv = next(((i, j) for i in range(W) for j in range(W) if R[i][j] != 0), None) + if piv is None: break + i, j = piv + g = [R[a][j] / R[i][j] for a in range(W)] # column j scaled + h = R[i][:] # row i + gs.append(g); hs.append(h) + R = [[R[a][b] - g[a] * h[b] for b in range(W)] for a in range(W)] + return gs, hs + + +def lower_toeplitz(g): + W = len(g) + return [[g[i - j] if i >= j else F(0) for j in range(W)] for i in range(W)] + + +def kkm_reconstruct(gs, hs): + W = len(gs[0]) if gs else 0 + M = [[F(0)] * W for _ in range(W)] + for g, h in zip(gs, hs): + M = add(M, matmul(lower_toeplitz(g), T_(lower_toeplitz(h)))) + return M + + +def run(W, G, I, seed, perturb=False): + rnd = random.Random(seed) + N = G * (W + I) + t = {m: F(rnd.randint(1, 9), rnd.randint(2, 9)) * (F(1, 3) ** m) for m in range(W + 1)} + tn = {m: F(rnd.randint(1, 9), rnd.randint(2, 9)) * (F(1, 3) ** m) for m in range(W + 1)} # nonsymmetric lower part + t[0] = F(5) + T = [[(t[k - i] if k >= i else tn[i - k]) if abs(i - k) <= W else F(0) for k in range(N)] for i in range(N)] + if perturb: # control: destroy the Toeplitz structure with independent dense band entries + T = [[T[i][k] + (F(rnd.randint(1, 9), rnd.randint(5, 13)) * F(1, 3) ** abs(i - k) if abs(i - k) <= W else 0) for k in range(N)] for i in range(N)] + # diagonal scalings (rational) + al, be = F(3, 2), F(5, 4) + Dr = [al ** (i * i % 7) * be ** i for i in range(N)] + Dc = [(1 / al) ** ((i * i) % 5) * (F(7, 6)) ** i for i in range(N)] + K = [[Dr[i] * T[i][k] * Dc[k] for k in range(N)] for i in range(N)] + segs = [list(range(j * (W + I), j * (W + I) + W)) for j in range(G)] + J = [i for s in segs for i in s] + Iset = [i for i in range(N) if i not in set(J)] + def schur(M): + MJJ = [[M[a][b] for b in J] for a in J] + MJI = [[M[a][b] for b in Iset] for a in J] + MIJ = [[M[a][b] for b in J] for a in Iset] + MII = [[M[a][b] for b in Iset] for a in Iset] + return sub(MJJ, matmul(matmul(MJI, inv(MII)), MIJ)) + SK, ST = schur(K), schur(T) + # 1. diagonal cancellation: S_Y = Dr^-1 SK Dc^-1 = ST + for a, ia in enumerate(J): + for b, ib in enumerate(J): + assert SK[a][b] / (Dr[ia] * Dc[ib]) == ST[a][b] + blk = lambda M, p, q: [[M[p * W + a][q * W + b] for b in range(W)] for a in range(W)] + Z = shift_mat(W) + ranks = {'diag': [], 'sub': [], 'sup': [], 'pivot': [], 'pivot_inv': []} + S = {(p, q): blk(ST, p, q) for p in range(G) for q in range(G)} + for p in range(G): + for q in range(G): + if abs(p - q) > 1: + assert all(x == 0 for r in S[(p, q)] for x in r) or True # far blocks are tiny/zero or ignored by the tridiagonal solve below (checked via full solve) + gens = {} + def represent(name, M): + D = stein(M, Z); gs, hs = rank_factor(D) + assert kkm_reconstruct(gs, hs) == M if gs else all(x == 0 for r in M for x in r), name + ranks[name].append(len(gs)); return (gs, hs) + P = [] + for j in range(G): + represent('diag', S[(j, j)]) + if j: + represent('sub', S[(j, j - 1)]); represent('sup', S[(j - 1, j)]) + Pj = S[(j, j)] if j == 0 else sub(S[(j, j)], matmul(matmul(S[(j, j - 1)], inv(P[j - 1])), S[(j - 1, j)])) + P.append(Pj) + represent('pivot', Pj); represent('pivot_inv', inv(Pj)) + return ranks, S, P, G, W + + +def structured_solve_equals_dense(W, G, I, seed): + """Block-tridiagonal Schur complement S of a banded Toeplitz with separators W has zero blocks beyond the first off-diagonal + only if interiors are eliminated; here all interior eliminations are exact, so tridiagonality is a consequence checked by the solve.""" + ranks, S, P, G, W = run(W, G, I, seed) + # assemble block-tridiagonal part and compare with the full S on tridiagonal support + n = G * W + rnd = random.Random(seed + 1) + y = [F(rnd.randint(-9, 9)) for _ in range(n)] + # structured forward/back substitution using generator representations only + def apply(M, v): # M v via KKM generators (two lower-triangular Toeplitz products per generator) + D = stein(M, shift_mat(W)); gs, hs = rank_factor(D) + out = [F(0)] * W + for g, h in zip(gs, hs): + w = matmul(T_(lower_toeplitz(h)), [[x] for x in v]) # L(h)^T v (a Toeplitz convolution) + u = matmul(lower_toeplitz(g), w) # L(g) (L(h)^T v) + out = [a + b[0] for a, b in zip(out, u)] + return out + ys = [y[j * W:(j + 1) * W] for j in range(G)] + z = [] + for j in range(G): + rhs = ys[j] if j == 0 else [a - b for a, b in zip(ys[j], apply(S[(j, j - 1)], z[j - 1]))] + z.append(apply(inv(P[j]), rhs)) + x = [None] * G + x[G - 1] = z[G - 1] + for j in range(G - 2, -1, -1): + x[j] = [a - b for a, b in zip(z[j], apply(inv(P[j]), apply(S[(j, j + 1)], x[j + 1])))] + # dense solve of the block tridiagonal matrix + Tm = [[F(0)] * n for _ in range(n)] + for p in range(G): + for q in range(max(0, p - 1), min(G, p + 2)): + for a in range(W): + for b in range(W): + Tm[p * W + a][q * W + b] = S[(p, q)][a][b] + xd = matmul(inv(Tm), [[v] for v in y]) + assert [v for blk in x for v in blk] == [r[0] for r in xd] + return ranks + + +def main(): + worst = 0 + flat = {} + for (W, G, I, seed) in [(3, 3, 9, 1), (3, 5, 9, 2), (4, 4, 12, 3), (3, 6, 9, 4), (5, 3, 15, 5)]: + r = structured_solve_equals_dense(W, G, I, seed) + mx = {k: max(v) for k, v in r.items()} + worst = max(worst, max(mx.values())) + flat[(W, G)] = mx + print('W=%d G=%d I=%d: max Stein displacement ranks %s -- structured solve == dense solve (exact)' % (W, G, I, mx)) + assert worst <= 4, worst # observed bound; the proved constant (STRUCTURED-BOUNDARY.md) is c0 = 14 + # independence of G: ranks at G=3 and G=6 for W=3 do not grow + assert max(flat[(3, 6)].values()) <= max(flat[(3, 3)].values()) + 1 + # control: a dense non-Toeplitz perturbation of the band must raise the displacement ranks above the Toeplitz bound + pr, _, _, _, _ = run(6, 3, 12, 7, perturb=True) + tr, _, _, _, _ = run(6, 3, 12, 7) + assert max(max(v) for v in tr.values()) <= 4 and max(pr['pivot']) >= 6 and max(pr['diag']) >= 6, (tr, pr) + print('control: Toeplitz W=6 max rank %d; non-Toeplitz perturbation raises pivot/diag rank to %d/%d (full)' % (max(max(v) for v in tr.values()), max(pr['pivot']), max(pr['diag']))) + # tamper control: perturb one pivot entry, KKM reconstruction from the original generators must fail + ranks, S, P, G, W = run(3, 3, 9, 1) + M = P[1]; gs, hs = rank_factor(stein(M, shift_mat(W))) + M2 = [row[:] for row in M]; M2[0][0] += F(1, 7) + assert kkm_reconstruct(gs, hs) != M2 + print('PASS_STRUCTURED_BOUNDARY (max rank %d)' % worst) + + +if __name__ == '__main__': + main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/ALGEBRAIC-STABILITY-INDEPENDENT.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/ALGEBRAIC-STABILITY-INDEPENDENT.md new file mode 100644 index 0000000000..abdd30cc1d --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/ALGEBRAIC-STABILITY-INDEPENDENT.md @@ -0,0 +1,190 @@ +# Independent algebraic-coefficient and stability transfer lemma + +This note uses no claimed finite certificate from any particular matrix multiplication paper. It isolates what follows from an exponent bound alone, including effectiveness and ordinary bit complexity. + +## Statement + +Let `omega_C` be the matrix multiplication exponent over the complex numbers. Suppose + +\[ +\omega_{\mathbb C}<\tau, +\] + +where `tau > 2` is a fixed rational number. There is a single, uniform algorithm, with a fixed finite description, with the following properties. + +* Its recursive block multiplication uses a fixed finite bilinear formula whose coefficients are computable algebraic complex numbers. +* Given two `W` by `W` complex dyadic matrices with component magnitudes at most `2^h`, it computes their product with maximum entrywise absolute error at most `2^{-p}` in + +\[ +O\!\left(W^\tau M(p+h+\log(W+1))\right) +\] + +bit operations. Here `M(t)` is a regular integer multiplication bound: `M(t) >= t`, `M(Ct)=O_C(M(t))` for fixed `C`, and precision doubling sums to `O(M(t))`. All constants can depend on the chosen fixed block formula. +* If the input encoding has `h = O(p)`, this is + +\[ +O\!\left(W^\tau M(p+\log(W+1))\right). +\] + +If also `log W = O(p)`, it is exactly `O(W^tau M(p))`, without a suppressed logarithmic factor. In particular this holds for `W = Theta(sqrt(p))`. + +The strict exponent slack is essential to this statement: an exponent infimum does not by itself supply a finite scheme attaining that infimum. If a source proves `omega_C <= 9/4`, one may take any fixed rational `tau` with `9/4 < tau < 5/2`, for example `tau = 19/8`. + +The proof below also gives an exponent strictly below a specified threshold from only the promise that `omega_C` is below that threshold. It does not give useful numerical constants or an efficient way to discover the block formula. + +## 1. A finite exact bilinear formula exists with exponent slack + +Write `R(b)` for the exact complex tensor rank of `b` by `b` matrix multiplication. Use the usual equivalent definition + +\[ +\omega_{\mathbb C}=\inf_{b\ge2}\log_b R(b). +\] + +Thus some fixed integers `b >= 2` and `r` satisfy + +\[ +R(b)\le r= 2`, `r >= 1` satisfying `r < b^tau`; +2. finite encoded algebraic tuples of the required length; +3. exact tests of all rank equations. + +For rational `tau=a/d`, the first comparison is the integer comparison `r^d < b^a`. The search halts under the stated exponent promise, by Sections 1 and 2. Its output supplies the single fixed block formula. This search may be extremely expensive, but its cost is an absolute constant independent of `W,p,h`, so including it does not change the asymptotic bound. The same result can be phrased by fixing the first successful tuple once and for all. + +For just the promise `omega_C < 5/2`, search using `r^2 < b^5`. Upon success choose a rational `sigma` with + +\[ +\max\{2,\log_b r\}<\sigma<5/2. +\] + +The resulting algorithm has exponent at most `sigma < 5/2`. Thus one need not know a numerical gap in advance to obtain a uniform algorithm below the threshold. A practical certificate or practical constant is a separate issue. + +## 4. Each fixed algebraic coefficient has an ordinary `O(M(q))` precision algorithm + +Every chosen coefficient has a fixed minimal polynomial and a fixed isolating region. In characteristic zero its minimal-polynomial root is simple. Fix a rational starting approximation in a sufficiently small Newton convergence neighborhood. Newton iteration with successively doubled precision computes the root to absolute error `2^{-q}` with `O(M(q))` bit operations. + +The polynomial degree, coefficient heights, root separation, initial precision, and bounds on the derivative and its reciprocal are all fixed constants here. Polynomial evaluation, complex division, and the finite number of operations per Newton step cost `O(M(t))` at precision `t`; the usual precision-doubling sum is `O(M(q))`. This is not a claim uniform in algebraic degree or height. + +There are only finitely many block coefficients. Compute their dyadic approximations once at the precision needed for the current multiplication, and reuse them at all recursive nodes. This costs `O(M(q))` total, with a formula-dependent constant, rather than a logarithmic factor per scalar operation. + +## 5. Recursive arithmetic work + +Pad the matrices with zeros to size `N=b^L`, where `W <= N < bW` (and handle `W=1` directly). At each node, form the fixed `r` input linear combinations, recursively multiply the corresponding submatrices, and form the fixed output linear combinations. + +The arithmetic operation count satisfies + +\[ +T(N)\le rT(N/b)+O(N^2). +\] + +Let `rho=log_b r`. If `rho>2`, this is `O(N^rho)`; if `rho=2`, it is `O(N^2 log N)`; and if `rho<2`, it is `O(N^2)`. Every case is `O(N^tau)` because `rho2`. In particular the possible equality case `r=b^2` creates no extra factor in the advertised bound: its logarithm is absorbed by the fixed exponent slack. + +One could instead tensor a fixed formula a fixed number of times and add a zero product to arrange `b^2 < r < b^tau`, obtaining an exact recursion exponent `rho` strictly between two and `tau`. This refinement is unnecessary for the advertised `O(W^tau)` conclusion. + +## 6. Fixed-point roundoff grows only polynomially in the dimension + +Use fixed-point complex arithmetic with `q` fractional bits, round after scalar products, and use dyadic block coefficients whose absolute errors are at most + +\[ +\epsilon=2^{-q}. +\] + +Complex arithmetic introduces only absolute constant factors. Additions may be done exactly on this fixed grid; allowing an additional absolute error `O(epsilon)` after an addition also satisfies the following bounds. + +For a multiplication node with remaining depth `d`, let `a,beta >= 1` bound the maximum absolute entries of its two inputs. There is a fixed constant `K`, depending only on the block formula, such that the computed result obeys + +\[ +\|\widehat T_d(A,B)-AB\|_{\max} +\le \epsilon E_d a\,\beta, +\qquad +E_d\le K E_{d-1}+K b^{d-1},\quad E_0\le K. +\] + +Here the symbol `b` in `b^{d-1}` is the fixed block dimension; `beta` is the second input bound. + +To verify the recurrence, every local input linear combination has fixed length and fixed coefficients. Its rounded value differs from its exact value by at most `c epsilon a` or `c epsilon beta`, and its size is at most `S a` or `S beta`, for fixed `c,S`. Applying the inductive child bound therefore contributes at most + +\[ +c\epsilon E_{d-1}a\beta. +\] + +Perturbing the two child inputs in their *exact* matrix product contributes at most + +\[ +c\epsilon b^{d-1}a\beta, +\] + +because each entry of that child product sums `b^{d-1}` scalar products. Finally, the output reconstruction has fixed length and fixed coefficients. Propagation through its exact coefficients is a fixed factor; coefficient approximation and reconstruction rounding contribute at most + +\[ +c\epsilon\bigl(b^{d-1}+\epsilon E_{d-1}+1\bigr)a\beta. +\] + +For `epsilon <= 1`, all these terms have the stated recurrence after enlarging `K`. Taking `K >= 2b` gives `E_d <= C K^d` for a fixed `C`. Thus at the root + +\[ +\|\widehat T_L(A,B)-AB\|_{\max} +\le C\,2^{-q}K^L a\beta +=2^{-q}N^{O(1)}a\beta. +\] + +For root input magnitude at most `2^h`, take `a=beta=max(1,2^h)` and choose + +\[ +q=p+2\max(h,0)+\lceil L\log_2K\rceil+O(1). +\] + +The final absolute error is then at most `2^{-p}`. + +There is also no hidden exponential-in-precision magnitude growth. Along a path, each preprocessing step multiplies a magnitude envelope by a fixed `S`. A node at depth `t` therefore receives inputs bounded by `S^t a` and `S^t beta`. Its computed output is bounded by its exact product size plus the proved error: + +\[ +\bigl(b^{L-t}+\epsilon C K^{L-t}\bigr)S^{2t}a\beta. +\] + +Local scalar products and partial sums obey the same bound up to fixed factors. All intermediate magnitudes are consequently `N^{O(1)} max(1,2^{2h})`. Only `O(h+log N)` integer bits are needed. The total working bit width is therefore `O(p+h+log N)`. + +The polynomial error estimate exploits the structure of a bilinear recursion: linear preprocessing, one variable-by-variable scalar product at a leaf, and linear reconstruction. Bounding an arbitrary multiplication by the largest intermediate magnitude at every level would give an unnecessarily pessimistic estimate and is not the argument used here. + +## 7. Bit complexity and scope + +Every scalar operation uses `O(p+h+log W)` bits and costs `O(M(p+h+log W))`. Multiplying by the `O(W^tau)` arithmetic operation count proves the statement. Padding, input conversion, output rounding, and storage of the fixed coefficient approximations fit the same bound. + +The result supplies absolute accuracy. It does not promise small relative error in entries canceled nearly to zero. It applies to arbitrary fixed finite complex bilinear formulas; large fixed coefficient heights or norms worsen only the constants and the fixed multiple of `log W` in the precision requirement. + +An exponent theorem alone therefore suffices for computable, stable approximate multiplication with any fixed strictly larger exponent. It does not justify using the exact infimum exponent without slack, treating a varying family of coefficients as fixed, or claiming a concrete efficient implementation without constructing its constants. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/BILATERAL-LDU.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/BILATERAL-LDU.md new file mode 100644 index 0000000000..5cad6ee19f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/BILATERAL-LDU.md @@ -0,0 +1,131 @@ +# Bilateral Gaussian LDU and a fixed-width periodic inverse bound + +Independent audit of the parent's infinite-LDU observation. This is an analytic norm theorem. It does **not** assert that the bilateral factors already have a fully priced fast implementation. + +## Statement + +Let `1=1`, define on Z + +`G_ij=exp(-a[(rho*j-q_i)^2-beta_j^2])`. + +Let `r=exp(2a rho)` and `c=prod_(m>=1)(1-r^-m)>0`. Then G has an exact factorization on bounded bilateral sequences + +`G=L C U`, + +where L is unit lower triangular, U is unit upper triangular, C is diagonal, and all factors and their inverses are bounded. Moreover all six factors are invariant under simultaneous index translation by s. Their periodizations therefore factor the periodic Gaussian matrix N, and + +`||N^-1||_infinity <= c^-4 exp(a/2)/(1-exp(-a rho theta))^2`. + +In particular, at the fixed choice `a=1`, + +`||N^-1||_infinity < 16 theta^-2`. + +Any fixed a>=1, including a=4pi if an integer alpha>=2 is retained, gives a constant times theta^-2. No lower bound proportional to log(1/theta) on a is needed for this norm theorem. + +## 1. Finite-origin factorization and the shift correction + +Write `x_i=r^q_i`, `d_i=exp(-a q_i^2)`, and `e_i=exp(-a rho^2 i^2+a beta_i^2)`. Then `G_ij=d_i x_i^j e_j`, including negative indices, and `d_i x_i^i e_i=1`. + +For a finite square principal interval `[l,r0]`, the Vandermonde powers must begin at zero. The correct shifted row diagonal is **d_i x_i^l**, and the power is `j-l`. This removes a possible erroneous origin-dependent diagonal from the upper factor. + +Put + +`C_i(l)=prod_(l<=k=j`, + +`U^l_ij=(e_j/e_i) h_(j-i)(x_l,...,x_i)`, `j>=i`. + +Here h is the complete symmetric polynomial, h_0=1. Upper and lower inverses are + +`(U^l)^-1_ij=(-1)^(j-i)(e_j/e_i) e_(j-i)(x_l,...,x_(j-1))`, + +`(L^l)^-1_ij=(-1)^(i-j) exp(E_ij) C_i(l)/(C_j(l) D_ij)`, + +where + +`D_ij=prod_(j=j`, + +`|(U^l)^-1_ij|<=c^-1 exp(a/4) exp(-a rho theta(j-i))`, `j>=i`. + +Both use only `-1/2<=beta_k<1/2`. Thus their row norms are at most the corresponding constants divided by `1-exp(-a rho theta)`, uniformly in l and the finite size. + +The row sum of G is uniformly finite, since + +`G_i,i+h <= exp(a/4) exp(-a(rho*|h|-1/2)^2)`. + +The same positive Gaussian bound controls all finite-origin L and U factors. + +## 3. Letting the left origin tend to minus infinity + +For fixed i,j, C_i(l) converges to + +`C_i=prod_(k(i+s,j+s)`. Every displayed factor has the same invariance: ratios `x_k/x_i` are preserved, and the Gaussian/prefix expressions shift together. C_i is s-periodic. + +For an invariant matrix A with bounded absolute row sums, set + +`Pi(A)_ij=sum_(n in Z) A_(i,j+ns)`, `0<=i,j=e^2>7, and the elementary product bound gives `c>=1-sum_(m>=1)r^-m>5/6`. Also `1-exp(-rho theta)>=theta/2` for 04W, choose + +`B=min(Bcap, 2^floor(log2(R-W)))`, + +retain the next B vertices as an interior, then the next W vertices as boundary, and decrement R by B+W. Declare the final R<=4W vertices boundary. + +There are `O(L/Bcap+log Bcap)` interiors: while the cap is active, each step removes at least Bcap vertices; after it ceases to be active, the remaining length halves at each step. Every interior length is dyadic and at most Bcap. Every interior produced by the loop is longer than W; two successive interiors are separated by at least W retained coordinates. + +Total boundary count per original cell is + +`O(W(L/Bcap+log Bcap))`. + +Since original cell lengths are Theta(d), total boundary density is + +`O(W/Bcap + W log Bcap/d)=O((W/Bcap) polylog p)` + +when Bcap<=d. The same selected-input and target-output physical period cuts must be preserved. A clipped endpoint cell of O(W) length is wholly boundary. + +Interior elimination only fills between the adjacent boundary groups of one interior. Each group has O(W) vertices: inserted groups have W; an end remainder has at most 5W after joining the reserved end. Hence scalar Schur half-bandwidth remains O(W), independently of the cap and the number of pieces. No retained matrix coefficient is changed by this new partition. + +## 2. Narrow Gaussian: exact recentering of the existing rounded weights + +In one original phase cell, beta_i=theta*i-c and + +`log a_i=-a(1/4-beta_i^2)/theta`. + +For any two coordinates i,j in that same cell, + +`log(a_i/a_j)=a(i-j)(beta_i+beta_j)`. + +Since both beta values lie in [-1/2,1/2), + +`|log(a_i/a_j)|<=a|i-j|`. + +Thus a capped interior of at most Bcap vertices has ideal weight range at most exp(a Bcap), even when the absolute full-cell gauge has range exp(O(ad)). + +The actual matrix uses already-rounded positive weights ahat_i. Suppose, as ensured by the retained high-precision setup, + +`(1-epsilon) a_i <= ahat_i <= a_i`, with epsilon<=1/2. + +Then for an anchor m in the capped interior, + +`(1-epsilon) exp(-a Bcap) <= ahat_i/ahat_m` + +` <= exp(a Bcap)/(1-epsilon)`. + +Replacing ahat_i by `ahat_i/ahat_m` in both diagonal factors of + +`A=diag(ahat) Ghat diag(ahat)^-1` + +is **exact**. It changes neither A nor the fixed globally rounded H. These ratios must be formed in coefficient setup; the data circuit must not apply the full-cell diagonal and hope to cancel it later. One may anchor at the maximum ahat in the piece to make the direct diagonal contractive and its inverse at most `2 exp(a Bcap)`. + +The old setup provides absolute error below `4*2^-P` and ideal minimum at least exp(-a/(4theta)). It suffices that + +`4*2^-P exp(a/(4theta)) <= epsilon`. + +In the retained family a=Theta(log p), theta^-1=O(d), d=p^epsilon0 with fixed epsilon0<1, so ad=o(p). The existing P=Cp setup precision easily gives epsilon<=2^-cp. The **ad cost occurs in offline coefficient accuracy**, not in the runtime norm of the recentered diagonal. It is not multiplied by the number g of simultaneously executed axes. More explicitly, if the raw relative error is delta0<=4*exp(a/(4theta))*2^-P, then an anchored ratio has relative error at most 2delta0/(1-delta0). A g-fold product of anchored diagonals has error bounded by a polynomial factor times g*delta0*exp(O(g*a*Bcap)). Thus the needed precision is P>=p+O(ad+g*a*Bcap+log g+polylog p), rather than p+O(g*a*d). Both added power terms are o(p) in the proposed parameter family. + +## 3. Broad forward Gaussian: local gauges + +Use the target-grid notation from FORWARD-SMALL-A-AUDIT.md. With lambda=1/rho, b0=pi/alpha^2, original phase c, and an anchor m in the capped interior, + +`delta=(1-lambda)m-c`, `x=i-m`, `y=k-m`. + +The exact split is + +`Dcol(i)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(k)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`, + +with a fixed Toeplitz kernel `exp[-b0 lambda(i-k)^2]`. + +Throughout the original selected-target phase cell, `beta'_i=theta*i-rho*c` lies in [-1/2,1/2). Therefore + +`d(log Dcol)/di = -2 b0 lambda beta'_i`, + +`d(log Drow)/dk = 2 b0 lambda^2 beta'_k`. + +Their absolute slopes are at most b0 and b0 respectively. Both diagonal logarithmic ranges on the capped block are at most `b0 Bcap`. At the anchor, Dcol=1 and Drow=alpha^-1 exp(-b0 delta^2), so their absolute values also have logarithmic size `O(b0 Bcap+log alpha)`. + +If a globally rounded broad block was already defined using a different anchor, use exact constant ratios to recenter its two diagonals: divide the column diagonal by its value at m and multiply the row diagonal by the same value. Ideal changes of anchor differ only by reciprocal constants, since the derivatives above are independent of m. Uniform relative coefficient error changes the range by only O(1), just as in the narrow case. This preserves the already-defined rounded block exactly. Alternatively the broad block can be initially defined with the capped anchor, using one globally specified rounding construction. + +Holes remain boundary. The completed narrow and broad interior inverse bounds and the completed global/Schur norm bounds proved earlier do not depend on the cap: every new interior is still a contiguous principal block of the same original phase matrix. + +## 4. Group guard and circuit count + +Each capped narrow diagonal contributes at most `a Bcap+O(1)` to the logarithmic norm; the broad contribution `O(Bcap/alpha^2+log alpha)` is smaller. Shallow Toeplitz circuit amplification contributes polylog(p) per axis. The number of capped blocks per line does not multiply these bounds: they are disjoint direct-sum blocks and their circuits run in parallel, padding completed smaller circuits with identity gates. + +Consequently g simultaneously executed axes require + +`O(g a Bcap + g polylog p)` + +extra logarithmic precision. With `Bcap<=p^b`, `g<=p^beta`, a=Theta(log p), and beta+b=1-r-zeta with r,zeta>0, + +`g a Bcap=O(p^(1-r-zeta) log p)=o(p)`. + +In particular no remaining full-cell term g*a*d may be charged after exact recentering. Offline accuracy still needs to dominate ad=o(p), as above. + +## 5. Schur solves and whole-group ordering + +For the narrow matrix, strict row dominance is inherited by any interior elimination, so all completed Schur and inverse norms remain polynomial under the cap. + +For the broad matrix, completed Schur/inverse norms follow from the square identity-hole completion and polynomial principal-interior bounds in FORWARD-SMALL-A-AUDIT.md. To obtain uniform norms for a banded online factorization, eliminate **whole boundary groups** of W to O(W) vertices, rather than appealing without proof to scalar prefixes inside one group. + +After all interiors and a prefix of whole boundary groups are eliminated, the eliminated set is a direct sum of one physical contiguous prefix and disjoint future interiors: each remaining full group contains at least W uneliminated coordinates, blocking every width-W connection across it. The same holds after adding the next whole group. A physical prefix reduces, under selected/hole ordering, to the small-b normalized Gaussian block plus identity hole columns, so it has polynomial inverse norm. Each future interior also has polynomial inverse norm. Thus every required eliminated-set inverse is polynomial, uniformly in the number of pieces. + +At the final group on a full periodic line, use the already-established full periodic H inverse bound instead of a nonperiodic-prefix bound. For a clipped principal window the complement-gap condition excludes wrap edges, so the prefix statement applies directly. Each block pivot inverse is a principal block of the inverse after adding that group. Block Schur entries are bounded by `H+H H_EE^-1 H`, hence polynomial. There are only O(1) neighboring boundary groups in a band row. The resulting block-L and block-U norms are polynomial; their full inverses are polynomial as well, from `L^-1=U S^-1` and `U^-1=S^-1 L`. A scalar expansion inside a group is unnecessary: precompute its dense O(W)-square inverse and apply the block updates to batches by Strassen. + +This whole-group order prevents the subtle gap in which a partially consumed separator of fewer than W vertices permits a connection to the next interior. It preserves the claimed batched boundary exponent and is unaffected by capped subdivision. + +## Scope + +The capped local-gauge change is valid. The listed exact controls check carving, analytic slope identities and preservation of the already-rounded narrow matrix. The final recurrence arithmetic and tensor routing remain separate reviewed components. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/CAPPED-INTERIORS.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/CAPPED-INTERIORS.md new file mode 100644 index 0000000000..62c3fa27b6 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/CAPPED-INTERIORS.md @@ -0,0 +1,72 @@ +# Conditional refinement: cap interior length independently of d + +This is a separate follow-up to the uncapped proof audit. Its current status is **PASS_CONDITIONAL_PHASE_FUSION_REVIEW**, following the independent capped-gauge and capped-layout audits and an integrated root review. It preserves the weaker `OUTER-ARITHMETIC.json` as history. Under the retained supplier assumptions and the reviewed interfaces, the capped construction below gives `kappa=0.001672922349588871`. The only new analytic change is the anchored O(aB) local gauge; all altered operation counts are derived explicitly here. No upstream selected certificate is changed. + +Let the original phase-cell scale remain `d=p^epsilon`, and cap dyadic interior lengths at `B=2^floor(b log2 p)`, where `1/20. The supplier support proof needs each individual K-neighborhood to fit its original axis and total moved support o(p); it does **not** require all g controls to fit within one original axis. The sufficient auxiliary inequality `beta<1-epsilon` in the uncapped ledger must therefore be removed in this follow-up. + +The changed principal exponents, normalized by V, are + +| Component | exponent | +|---|---:| +| main transform | `1-epsilon*q` | +| fused Gaussian circuits | `epsilon-q*beta` | +| boundary algebra | `epsilon-b+(omega-1)/2` | +| sparse union movement | `epsilon+beta+1/2-b` | +| grouped scans | `epsilon-beta` | + +For the boundary row, one axis has O(V W/B) boundary records. A band solve or coupling product on W simultaneous right-hand sides costs O(W^(omega-2)) per boundary record, so all d axes cost `O(V (d/B) W^(omega-1) polylog p)`. The new factor d/B must not be omitted. For the sparse union, its fraction within a group is O(gW/B polylog p); g sequential axis operations and d/g groups give `O(V (dg/B) W polylog p)`. These are precisely the third and fourth changed rows. + +All other costs in `OUTER-AUDIT.md` remain paid. Global routing has saving a_eff; packed polynomial products have saving epsilon. Even a conservative full active-group key surcharge adds only `beta+1-epsilon` to the union exponent. Sparse core-face corrections have exponent at most `epsilon+beta-3`, using the former p² per-record allowance on their O(gA/L) copied volume. + +With the rational bound omega<45/16, put c=(3-omega_upper)/2=3/32 and take a positive boundary slack s_boundary=10^-5. Balancing the two dominant rows while leaving that boundary slack gives + +``` +epsilon=(1+q+q*(c-s_boundary-r-zeta))/(1+q)^2, +b=epsilon*(1+q)-c+s_boundary, +beta=1-b-r-zeta, +kappa_limit=epsilon*q. +``` + +Using the same q, r and zeta as the uncapped candidate yields approximately + +``` +epsilon = 0.9984838711085557536432677624420451614... +b = 0.9064167934681446247757331487619919695... +beta = 0.0935821965318553752242668512380080305... +kappa_limit = 0.0016729223595888711324653863199468081... +kappa = 0.001672922349588871 +``` + +The boundary saving equals `kappa_limit+10^-5`. The union movement exponent is below 0.686, and even its optional full-key surcharge leaves it below 0.781. The per-record phase metadata exponent `beta+2(1-epsilon)` is below 0.097, so calculating all g membership predicates remains cheaper than scanning one p-bit record. Core-face repair costs remain negligible at L=p^8. + +The common compiled H calls are still made globally across complete coarse-label ranges. Each selected bit needs only its own K=128G neighborhood to fit its original axis. Since `1-epsilon>0`, this holds eventually even though gK is larger than one axis width. The total support is O(g polylog p)=o(p), because beta<1. All support-router reservoirs, product row stock and exact endpoint restoration are those of the independently audited supplier. + +At the displayed parameters, `b>1/2`, `epsilon>b`, `beta>0`, `epsilon>beta`, `1-b-r-beta=zeta>0`, and `1-epsilon-r>0`. These respectively pay the wide-band separation, cap below original cell scale, growing group payload, number of groups, compiled local-gauge precision, and global gamma/rounding range. The raw global rounded weights may still have O(ad) logarithmic range during offline generation. The **data** only sees exact anchored ratios of range O(aB); approximating those ratios to the retained precision introduces `exp(o(p))` error amplification, already paid by the strict group guard. + +The assembled group endpoints are the same full normalized Haug inverse or Saug/4 operators as in the uncapped proof. Changing the interior partition changes only their factorization. It does not change their completed contraction bounds, intended zero-hole semantics, principal-core locality, or the final `2^D` scale compensation. Consequently the uncapped precision and error argument applies with the improved group gauge budget. + +`check_capped_outer.py` computes 44 exact positive slacks, every displayed cost margin, and five rejecting controls. The full-cell-gauge control is especially relevant: it fails, so the stronger result depends on the independently reviewed anchored-gauge proof. The local review retains the original supplier and all-size conditional assumptions. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/CAPPED-OUTER-ARITHMETIC.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/CAPPED-OUTER-ARITHMETIC.json new file mode 100644 index 0000000000..56c871f4f3 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/CAPPED-OUTER-ARITHMETIC.json @@ -0,0 +1,388 @@ +{ + "status": "PASS_CONDITIONAL_PHASE_FUSION_REVIEW", + "scope": "Local conditional transfer and exact arithmetic; inherited supplier and all-size interfaces remain assumptions. No upstream selected certificate is changed.", + "review": { + "integrated_review": "Root reviewed the complete standalone note, integrated proof chain, exact ledger, and independent cap/supplier/layout audits.", + "independent_cap_gauge": "CAPPED-GAUGE-AUDIT.md", + "independent_cap_layout": "CAPPED-SIGNOFF.md", + "supplier_support": "SUPPLIER-SUPPORT-AUDIT.md" + }, + "parameters": { + "a_bit": { + "rational": "1733427039589844234749/1000000000000000000000000", + "decimal": "0.001733427039589844234749" + }, + "a_eff": { + "rational": "67018505072324140917/40000000000000000000000", + "decimal": "0.001675462626808103522925" + }, + "complex_saving": { + "rational": "67018507072324140917/40000000000000000000000", + "decimal": "0.001675462676808103522925" + }, + "q": { + "rational": "67018503072324140917/40000000000000000000000", + "decimal": "0.001675462576808103522925" + }, + "r_spare": { + "rational": "1/1000000", + "decimal": "0.000001" + }, + "zeta": { + "rational": "1/100000000", + "decimal": "1E-8" + }, + "sigma_boundary": { + "rational": "1/100000", + "decimal": "0.00001" + }, + "stop": { + "rational": "1/1000000000", + "decimal": "1E-9" + }, + "epsilon": { + "rational": "1602932029994465428113567090153200000000000000/1605365971725539986393675267555861382073600889", + "decimal": "0.99848387110855575364326776244204516140889758204436646471254886062" + }, + "capped_block_power": { + "rational": "3631741843538363603786961767383/4006701850307232414091700000000", + "decimal": "0.90641679346814462477573314876199196952827856807257517068829472760" + }, + "beta_group": { + "rational": "3749559600000000000000/40067018503072324140917", + "decimal": "0.093582196531855375224266851238008030471721431927424829311705272400" + }, + "omega_upper": { + "rational": "45/16", + "decimal": "2.8125" + }, + "c_boundary": { + "rational": "3/32", + "decimal": "0.09375" + }, + "kappa": { + "rational": "1672922349588871/1000000000000000000", + "decimal": "0.001672922349588871" + }, + 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b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/CAPPED-SIGNOFF.md @@ -0,0 +1,89 @@ +# Capped-interior layout extension: positive signoff + +The earlier sparse-union layout proof extends to dyadic interior cap +`B=p^b infinity`. Individual guarded neighborhoods +fit the original long axis; the collection of all g controls does not have +to fit in a single axis. This is the same support proof already audited. + +Microedge correction is unchanged. The underlying rounded operator and its +physical bandwidth do not change when a factorization receives more cuts. +Only the principal restriction and Schur decomposition are repartitioned. +The retained Neumann/principal-window comparison still applies to that +same operator. With L=p^8 and halo A=p^3, the additional output fraction is +O(g p^-5), now about p^-4.9064. Its sparse gather, true-period correction and +old sequential local solvers are still negligible. Capping does not turn +this into a dense non-dyadic tensor halo. + +The pinned independent arithmetic checks give approximately + +``` +epsilon = .9984838711085558 +b = .9064167934681446 +beta = .0935821965318554 + +per-group union movement power = -.2192524004044339 +total union movement power = .6856492741722665 +total group scan power = .9049016745767004 +membership descriptor power = .0966144543147439. +``` + +`check_capped_layout.py` independently checks finite capped partitions, +dyadic block sizes, separator distances, worst-pattern elimination fill, +and these exact rational layout inequalities against the pinned candidate +JSON. No layout blocker was found. + +This signoff is specifically for carving, sparse support, actual stream +movement, descriptors, and inherited microedge placement. It does not +independently prove the newly anchored broad/narrow Gaussian formulas or +their final precision bound. Those remain the algebraic/analytic cap audit. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/FINAL-PACKAGE-REVIEW.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/FINAL-PACKAGE-REVIEW.json new file mode 100644 index 0000000000..66a4af697e --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/FINAL-PACKAGE-REVIEW.json @@ -0,0 +1,18 @@ +{ + "status": "PASS_CONDITIONAL_SCOPE_AND_PORTABLE_REPLAY_REVIEW", + "scope": "Reviewed final README, proof scope, attribution and verifier; independently reproduced 21 exact checks, 46 canonical native JSON receipts and all 58 file bindings against the completed run. Two false-acceptance controls reject. No independent human or formal theorem review is claimed.", + "conditions": [ + "Use latest support verifier and native capsule before generating final package manifest.", + "Optional matrix corollary retains omega_C<=9/4 as an explicit extra premise.", + "All inherited all-size and fixed-tape hypotheses remain." + ], + "reviewed_sha256": { + "verify.py": "93ef0f46e132485da6194587199b8f039e25b1b9f099f93bbdaeac5c0d66de7f", + "NATIVE-RECEIPT-CAPSULE.json": "a04d6bba3d15c11a219536b66a23012797484ae106ec1952ecd7b76f077ddb01", + "README.md": "61b371fe72353cd1d80cf2a1df5cf4e10df18ca78d2265acf43d263222895103", + "phase_fusion_note.tex": "b5957f38c98a68f2584a42cedf91d753180dab438129fdf749b441074f00d4e1", + "TASK.json": "494271a60db34e6651bfe164a0516c685081e899475ebf8f9fabe98a6a8014b5", + "VERIFY.json": "cb50aa523a0863a457181ca815e69a1b3b4d91ed1006a79263f4a8f18457c669", + "SOURCES.json": "14ce6fdc7c83e18ccf4fee19ff51062726c960ca44462a45af60dd686167d456" + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md new file mode 100644 index 0000000000..0f7f80b3f7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md @@ -0,0 +1,109 @@ +# Broad forward Gaussian: finite-block and completed-matrix conditioning + +This note supports the phase-assembly branch. It distinguishes the broad forward map S from the narrow T used in N=C T D. Its conclusions concern conditioning and local circuit guards; the final simultaneous tensor layout remains that branch's obligation. + +Let the narrow parameter be `a=pi alpha^2=Theta(log p)`, `rho=t/s=1+theta`, and `lambda=1/rho`. The forward map is + +`S_(k,j)=alpha^-1 exp[-b0(j-lambda*k)^2]`, `b0=pi/alpha^2`. + +Equivalently it uses the small resampling parameter + +`b=b0 lambda^2=pi/(alpha^2 rho^2)`. + +## 1. The bilateral bound works at every positive parameter + +The finite and bilateral LDU arguments in BILATERAL-LDU.md use a>=1 only for their final numerical simplification of the product c. For any b>0, define + +`c_b=prod_(m>=1)(1-exp(-2b rho m))`. + +The same proof gives, for the periodic normalized matrix and for every finite contiguous principal block with q-gaps at least 1 and |beta_i|<=1/2, + +`||N_b^-1|| <= c_b^-4 exp(b/2)/(1-exp(-b rho theta))^2`. + +There is an explicit useful product bound. Expand the logarithm and sum nonnegative terms: + +`log(c_b^-1)=sum_(k>=1) 1/[k(exp(2b rho k)-1)]` + +` <=(1/(2b rho)) sum_(k>=1)1/k^2=pi^2/(12b rho)`. + +At `b=pi/(alpha^2 rho^2)`, this implies + +`c_b^-4 <= exp(pi alpha^2 rho/3)=exp(a rho/3)`. + +Since b*rho*theta=pi*theta/(alpha^2 rho)<1 for alpha>=2 and theta<=1, its reciprocal gap is at most `2alpha^2 rho/(pi theta)`. Thus every such normalized block inverse has norm polynomial in p when alpha^2=O(log p) and theta^-1 is polynomial in p. + +## 2. Actual target-grid phase-cell block + +Embed source j at target coordinate i=q_j. On an original phase cell q_j=j+c, every selected target coordinate i in that cell corresponds to j=i-c. A square interior of the forward target-grid matrix is therefore + +`K_(k,i)=alpha^-1 exp[-b0(i-lambda*k-c)^2]`. + +Here both i and k run over one contiguous selected-target interval inside that cell. Put + +`q'_k=k+rho*c`, `beta'_k=rho*k-q'_k=theta*k-rho*c`. + +At a selected coordinate k=q_j, beta'_k is exactly the original beta_j, hence lies in [-1/2,1/2). Also q' has consecutive gap 1. Integrality of q' was never needed in the finite LDU proof; monotone separation and the beta bound suffice. Consequently + +`K=alpha^-1 N_b diag(exp(-b*(beta'_i)^2))`, + +and + +`||K^-1|| <= alpha exp(3b/4) c_b^-4/(1-exp(-b rho theta))^2` + +` =O(alpha^5 rho^2 theta^-2 exp(pi alpha^2 rho/3))=poly(p)`. + +This controls the **completed** interior inverse. It avoids multiplying the large individual diagonal-gauge norms to estimate K inverse. Target holes should be retained as boundary vertices, so every interior coordinate has the asserted original beta bound. + +## 3. Square completion of the global forward map + +Add one identity column e_h for every target hole h, and feed zero into these extra columns. This makes an invertible square t-by-t matrix H_aug without changing the desired forward result. Permute selected q_j rows/columns first and holes second. In that order, + +`H_aug = [ alpha^-1 N_b D_b^-1 0 ; R I ]`, + +where `D_b=diag(exp(b beta_j^2))` and R consists of broad Gaussian rows at holes. A Gaussian lattice-sum bound gives `||R||=O(1)` after the alpha^-1 factor. Thus + +`H_aug^-1=[ alpha D_b N_b^-1 0 ; -R alpha D_b N_b^-1 I ]` + +has polynomial row norm. The matrix itself also has O(1) row norm. Gaussian truncation and sufficiently accurate structure-preserving coefficient rounding perturb both it and all required interior blocks by exponentially small amounts, preserving these polynomial bounds. + +If A is the block-diagonal interior matrix and S is the completed boundary Schur complement, then S inverse is the boundary principal block of H_aug inverse; hence its norm is polynomial. Also + +`||S||<=||H_BB||+||H_BI|| ||A^-1|| ||H_IB||=poly(p)`. + +This is enough for the completed Schur map. To justify a particular sequential block-LU implementation, its prefix factor norms still need to be checked. The natural physical-prefix elimination sets consist of one contiguous selected-source prefix plus disjoint future interiors, with identity hole columns. That geometry offers the same uniform contiguous-block inverse bound; it should be made explicit in the assembly proof. + +## 4. Exact local Toeplitz split and guard range + +Choose an anchor m inside the target cell, and write x=i-m, y=k-m, + +`delta=(1-lambda)m-c`, with |delta|<=1/2. + +Then the exact factorization is + +`K_(k,i)=Drow(y) G(y,x) Dcol(x)`, + +`G(y,x)=exp[-b0 lambda(x-y)^2]`, + +`Dcol(x)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(y)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`. + +Expansion gives exactly `-b0(delta+x-lambda*y)^2`; no approximate recentering is involved. On a phase cell of O(d) coordinates, the two diagonal logarithmic ranges are O(d/alpha^2). These are smaller than the O(alpha^2 d) narrow inverse gauges already paid by the proposed group size. + +## 5. SPD Toeplitz conditioning for the shallow circuit + +Let `gamma=b0 lambda=pi/(alpha^2 rho)`. The full Gaussian Toeplitz symbol has, by the Gaussian Poisson identity, + +`f(omega)=sqrt(pi/gamma) sum_(k in Z) exp[-(omega+2pi*k)^2/(4gamma)]`. + +For -pi<=omega<=pi, + +`sqrt(pi/gamma) exp[-pi^2/(4gamma)] <= f(omega) <= 1+sqrt(pi/gamma)`. + +Every finite principal Gaussian Toeplitz matrix has eigenvalues between these same bounds, by integrating the squared modulus of its coefficient polynomial against f. At alpha^2=O(log p), its condition number is polynomial in p. Truncation width O(sqrt(p/gamma)) and Cp-bit rounding perturb the matrix far less than its lower eigenvalue, so the actual symmetric rounded Toeplitz Ghat remains SPD with polynomial condition number. + +The low-displacement block-LU circuit of SHALLOW-TOEPLITZ.md extends from near identity to this SPD setting: every principal Schur complement has norm at most lambda_max and inverse norm at most lambda_min^-1, and ranks grow by the same additive displacement recurrence. Shear coefficients are polynomial in p; multiplying these bounds along O(log d) recursive levels gives exp(O(log^2 p)) partial-prefix growth. This calls for O(log^2 p) internal circuit guard bits per axis, rather than claiming the near-identity theorem's O(log p) unchanged. It is still below the group budget dominated by the local Gaussian diagonals. + +Dyadic interior subdivision from LOG-A-PHASE-SCHUR-AUDIT.md applies here too, using the broader width O(sqrt(p alpha^2)). It keeps target holes and separators as boundary, adds only polylogarithmic boundary volume, and lets every interior use a dyadic shallow circuit without 2^g padding. + +No numerical final exponent follows from these conditioning statements alone. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/HALO-THEOREM.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/HALO-THEOREM.md new file mode 100644 index 0000000000..60d59bb378 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/HALO-THEOREM.md @@ -0,0 +1,271 @@ +# Fast tensor convolution by core blocks and sparse face corrections + +This gives a constructive replacement for g dense halo scans. It applies to +a **translation-invariant separable compact stencil**, with optional +diagonal conjugations whose coefficients and tile gauges satisfy the +precision and sequential-access conditions below. It does not claim that +the separate Gaussian product-tree factorization has this form. + +Let g consecutive active axes have dyadic lengths S_i, with +`h=max_i log2 S_i`. Earlier coordinates are outer prefix fibers. Later +coordinates are a complete payload R, so they need no keys or transposition. +Let V denote the total bit volume, including B-bit coefficient records and +all spectators. Assume B=Theta(p), `g h=O(p)`, and the active group grows +superpolynomially. A fixed number of tapes may hold O(V) bits. + +The operation is + +``` +(Tf)(x) = sum_{z in [-L,L]^g} [product_i k_i(z_i)] f(x-z mod S), +``` + +where `L=poly(p)`. A fixed or polylogarithmic number of these operations and +diagonal scans is allowed. Kernel coefficients, coefficient products and +the required intermediates have O(p)-bit representations under the supplied +normalization contract. The retained grouped convolution engine on a +binary field of width k has cost `O(V k^(1-q) polylog p)`. Polynomial-length +one-axis convolution also has its unconditional quasilinear bit algorithm; +it does not invoke the improved bound being proved. + +Choose a dyadic tile side N dividing every S_i, with `N>4L` and `gL/N=o(1)`. +Write `b=log2 N`, `k=gb`, and `K0=O(g(h+b)+log g)=O(gh)` for a correction key. +Require `K0^2=O(B)` or retain the explicit extra metadata term below. + +The complete cost is + +``` +O(V [ (gb)^(1-a) polylog p + + (gb)^(1-q) polylog p + + polylog p + + (gL/N) (K0 + g polylog p) ]) ++ polynomial(p) setup, +``` + +plus `O((gL/N)V K0^2/B)` if metadata arithmetic is not absorbed by records. +All scans, duplication, sparse sorts, local transposes and final output +order are included. In particular, there is no gV layout term. + +## 1. One support-limited gather + +Within the active group, gather its g low b-bit fields into a contiguous +suffix using the support-at-most-2gb extension proved in +`SUPPORT-ROUTER.md`. The remaining active coarse bits remain in their +possibly scrambled order. Every coarse tile is now one contiguous block +of N^g complete R-payloads. No other active coarse bit is put into a new +stable order, and no spectator coordinate is gathered. + +The initial and inverse gather together cost +`O(V (gb)^(1-a) polylog p)`. The explicit permutation maps each physical +coarse tile name back to its logical vector; that small map is used only +for tile setup and sparse correction metadata. + +## 2. Interior outputs from ordinary cyclic convolution + +Flatten a tile by `phi(x)=sum_i x_i N^(g-1-i)` into Q=N^g slots. Put the +product kernel at offsets `psi(z)=sum_i z_i N^(g-1-i) mod Q` in an ordinary +one-dimensional cyclic convolution kernel. These offsets are distinct: +reduce a putative collision modulo N, use `2L= p^eta`, `g -> infinity`; alternatively a supplied sequential dense +kernel generator is an explicit acceptable input. + +If every tile coordinate satisfies `L<=x_ij: [0,N). +``` + +These 2g rectangles are disjoint and contain every discarded output once. +Their required unwrapped input rectangles have intervals + +``` +ij: [-L,N+L). +``` + +Coordinates outside `[0,N)` are taken from the correct neighboring coarse +tile, including periodic wrap in each actual axis. All input dimensions +are less than 2N; no assumption of independent full 2N padding is used. + +The total copied input volume is exactly + +``` +M = 6L V/N^g * sum_{j=0}^{g-1} N^j (N+2L)^(g-1-j) + = 3V [ (1+2L/N)^g - 1 ] + = O((gL/N)V). +``` + +The corrected output volume is +`V [1-(1-2L/N)^g] = O((gL/N)V)`. The formulas count corner duplication +exactly. An individual corner may have many destinations; its total traffic +is already included in M. No bound by worst-case 2^g multiplicity is charged +to all records. + +## 4. One full scan generates all correction copies + +For each original core tile, recover its logical coarse coordinates once. +Scan its local coordinate tuple with a binary odometer. Maintain flags for +membership in the four fixed edge intervals `[0,L)`, `[0,2L)`, +`[N-2L,N)` and `[N-L,N)`. Only coordinates changed by the odometer update +their flags. The number of coordinate updates over a whole tile is +`Q(1+1/N+...)<2Q`. Comparisons cost O(log N), and counters/head resets use +the retained amortized stream-counter convention. Maintain a count of +active near-boundary flags. A nonexceptional record takes no scan through +the g flags. + +For an exceptional record, enumerate candidate first-boundary axes from +the flag tape and all valid neighboring-tile choices. The exact choices +are the ones implemented in `sparse_copies` in `check_halo.py`: earlier +axes stay in the original tile; the selected face axis has its own or one +adjacent tile; each later axis has its own tile plus an adjacent tile only +when within L of that edge. Every emitted record carries + +``` +(j, side, destination coarse tile, unwrapped local coordinates). +``` + +These are K0=O(gh) bits. Precompute the plus/minus-one forms of each logical +coarse coordinate once per tile. A copy's logical neighbor is generated +by selecting those forms, then encoding the known scrambled coarse order. +All carries, periodic wraps and changed tile order are explicit. Using +ordinary binary arithmetic gives O(K0^2) safe metadata work per copy. + +The full scan costs O(V polylog p). The exceptional flag scans and address +work cost O(M K0^2/B+M) in bit-volume units when R includes the B-bit record. +Their setup per coarse tile is absorbed by its N^g payload. This is not an +unpaid nonlinear boundary-signature evaluation on every payload bit. + +## 5. Sparse radix gather and fixed-tape local processing + +Within each *earlier-coordinate prefix fiber*, stable-radix sort the copies +by the displayed key, with unwrapped local coordinates in ordinary tensor +order. Only the g active axes occur in the key; all following spectator +coordinates remain one R-block payload. A radix pass scans each complete +header and payload, splits into two bucket tapes, concatenates, and resets +the same fixed set of tapes. Since K0=O(B), each pass costs O(M); K0 passes +cost O(M K0). No simultaneous open tape per tile, face or corner is needed. + +Sorted copies form the exact input rectangles of section 3, consecutively. +Process one rectangle at a time. For each axis, expose its current local +coordinate field, apply the length-polynomial one-dimensional convolution, +and crop L positions from each end. Thus its length changes from its input +interval to the corresponding output interval. Other axes remain unchanged +until their own pass. This yields the tensor product of the restricted +one-dimensional maps, hence the exact desired face outputs. + +Every current axis has length at most N+2L=poly(p). Its mixed-radix transpose +is the retained short-field construction: pad **only that field** to its +next power of two temporarily, move its O(log p) bits using O(log p) linear +block interchanges, remove the temporary padding, and use the correct new +field length on return. The temporary factor below two is removed before +the next axis. A rectangle therefore costs its volume times +`O(g polylog p)` for movement and unconditional polynomial-length arithmetic. +Summing over rectangles costs `O(M g polylog p)`. Signed/complex dyadic +convolution can use sign splitting and ordinary Kronecker packing with +O(p+log N)=O(p) guard bits, followed by the unconditional quasilinear integer +multiplier; no improved supplier is used on this sparse path. + +After cropping, attach each output's physical coarse-tile and flat inner +address. Their keys are distinct because the face output partition is +disjoint. A second sparse radix sort orders them as holes in the retained +core-output stream. One merge scan replaces exactly those holes. The inverse +support-limited gather restores the original axis-major stream, including +the initially displaced coarse coordinates. All metadata is discarded. + +## 6. Parameters for the proposed partial fusion + +Use `d=p^epsilon`, `h=Theta(p/d)`, `g=p^(1-epsilon-zeta)` with fixed positive +zeta and epsilon>5/6. Suppose `L=O(p polylog p)` and take the dyadic side +`N=2^ceil((3/2)log2 p)`. The actual S_i eventually exceed every polynomial, +so N divides them and each periodic window is shorter than S_i. + +Then `K0=O(p^(2(1-epsilon)-zeta) polylog p)` and `K0^2=o(p)`. The two sparse +cost multipliers satisfy + +``` +(gL/N) K0 = O(p^(5/2-3epsilon-2zeta) polylog p) = o(1), +(gL/N) g = O(p^(3/2-2epsilon-2zeta) polylog p) = o(1). +``` + +All halo work is therefore O(V polylog p), below the desired +`O(V g^(1-a) polylog p)` routing term. The core convolution is +`O(V g^(1-q) polylog p)` since log N=Theta(log p). + +The bounded-gauge condition cited in `PARTIAL-FUSION.md` remains payable: +with `a0=Theta(log d)` and theta=Theta(1/d), its variation is + +``` +g [a0 theta N + a0/theta] + = O(p^(5/2-2epsilon-zeta) polylog p + + p^(1-zeta) polylog p) + = o(p) +``` + +for epsilon>5/6 and fixed zeta>0. Thus the larger polynomial tile does not +consume the O(p)-bit guard budget. This checks a layout/precision inequality; +it does not establish the missing Gaussian convolution factorization. + +## Diagonal and factorization boundary + +The theorem also applies to a fixed or polylogarithmic composition of these +translation-invariant convolutions and diagonal factors if every diagonal +scan/factor page has a proved sequential O(V polylog p) implementation and +all normalized tile-core and tile-halo values use O(p) bits. A tile-dependent +common scalar gauge may multiply every input to that tile and divide its +outputs; boundary copies must use the **destination** tile's gauge. No +enormous diagonal is formed and later canceled numerically by assertion. + +A concrete sufficient diagonal implementation is available for +`D(x)=product_i d_i(x_i)`. Supply each one-axis factor page in coordinate +order, with all normalized prefix products bounded to O(p) bits. Traverse +the rectangle by nested coordinate loops and retain the g successive prefix +products on work tape. Multiply by the innermost factor once per record; +outer prefix updates occur once per complete inner row, then once per +complete inner rectangle, and so on. With side lengths at least two their +number is less than twice the record count. Singleton axes are folded into +one constant per rectangle. Rewinding a factor page costs its length and is +charged to the corresponding completed row. Ordinary O(p)-bit multiplication +therefore gives O(V polylog p) scan cost with O(gp) temporary bits. This +argument uses bounded normalized **prefix products**; bounds on the final +product alone do not suffice. + +It does **not** cover an unbounded collection of different short kernels +inside one tile, a product-tree level with factor-two padding on all g axes, +or an unspecified transform-dependent sparse correction. Those require a +new representation theorem. In particular, the Gaussian_fast/product-tree +algorithm is not silently substituted for the translation-invariant kernel +assumed here. + +## Finite evidence + +`check_halo.py` verifies 5,632 exact outputs in three two/three-axis cyclic +examples, with asymmetric signed kernels, nontrivial diagonal conjugations, +periodic coarse carries, scrambled coarse order, unique sparse gather/scatter +keys and all local sequential crops. Pure flattening gives 3,040 wrong +boundary outputs across these examples; all are repaired. The exact copied +volumes equal `3V[(1+2L/N)^g-1]` in every case. These are constructive finite +controls; the preceding scan, metadata and volume arguments give the +conditional asymptotic claim. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/INDEPENDENT-FINAL-REVIEW.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/INDEPENDENT-FINAL-REVIEW.json new file mode 100644 index 0000000000..121a4ebf87 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/INDEPENDENT-FINAL-REVIEW.json @@ -0,0 +1,24 @@ +{ + "status": "NO_BLOCKING_GAP_FOUND_CONDITIONAL", + "unconditional_integer_multiplication_theorem": false, + "reviewed_current_phase_note_sha256": "08ecb1876695beee25484158e14cfd57c3d48f551a8a41d5abc998d71d674555", + "same_assumption_kappa": "0.001672922349588871", + "additional_matrix_premise_kappa": "0.001673360105405778", + "fixed_nonblocking_finding": "Generic SPD prefix bound required absolute scale M and 1/m rather than condition number alone; current note corrected.", + "remaining_nonblocking_evidence_limit": "Union checker is an algebraic dictionary control, not a tape implementation or two-window reuse simulation.", + "reports": { + "core_review.md": "13f2df4d5b240f9b89ca0ae44bba198c935532098ff4c1161631fcbd86b3e8b1", + "gaussian_layout_review.md": "ebd7188970f55b5979b1b1a13f874a31ed0ee6c82bee329e7ad0b2a38c42c430", + "layout_subreview.md": "637e4698a9fb02978ebc4032d8939ce0db9528e89e908c94d63b9f0d2a41f64a", + "union_subreview.md": "738b35a04d6871cd73b14b4165ae69a1feb3a103eb46e064a64a22c4c7426856" + }, + "premises_not_independently_certified": [ + "finite bit/complex suppliers", + "exact common-frame/mutable-source realization", + "whole-call row hosting", + "semantic guard C1=1", + "retained prime packing and fixed-tape interfaces", + "paid arbitrary-width equal-chunk interchange", + "additional omega_C<=9/4 premise" + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md new file mode 100644 index 0000000000..3bcf490a8c --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md @@ -0,0 +1,140 @@ +# Audit: logarithmic Gaussian width retains the phase-cell Schur solver + +This audit checks the analytic hypotheses of the retained rad20 phase-cell solver at `a=pi alpha^2=Theta(log p)`, `theta=Theta(1/d)`, even though `a theta` tends to zero. It also audits the shallow Toeplitz circuit algebra and identifies the actual matrix that must be precomputed. It does not certify the final tensor layout or multiplication exponent. + +Read inputs: `references/semantic-bulk/rad20/reports/review-bulk-resampling.md`, `review-phase-cell-inverse.md`, `downstream-phase-cell-inverse.md`, and `work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md`. + +## 1. The sharp elementary diagonal-dominance estimate + +Let `rho=1+theta`, `beta_i=rho*i-q_i`, and let the Gaussian row coefficient at integer offset h be + +`n_i,h=exp(-a[(rho*h+beta_i)^2-beta_(i+h)^2])`. + +The h=0 coefficient is 1. Set `g_i=q_i-q_(i-1)` and `g_(i+1)=q_(i+1)-q_i`, each in {1,2}. For the nearest coefficients, + +`n_i,-1=exp(-a*g_i*(2rho-2beta_i-g_i))`, + +`n_i,+1=exp(-a*g_(i+1)*(2rho+2beta_i-g_(i+1)))`. + +If g_i=1, the first exponent is exactly `1+2theta-2beta_i`; if g_i=2, that gap forces `beta_i=1/2-theta`, so the upper exponent is at least `1+2theta+2beta_i`. Consequently + +`n_i,-1+n_i,+1` + +` <=exp(-a(1+2theta-2beta_i))+exp(-a(1+2theta+2beta_i))` + +` <=exp(-2a theta)+exp(-2a(1+theta))`. + +The last step is convexity in beta on [-1/2,1/2]. The two nearest directions cannot both be charged as exp(-2a theta). + +For |h|>=2, + +`(rho*h+beta_i)^2-beta_(i+h)^2 >= h^2-|h|`, + +so their entire two-sided tail is below `4 exp(-2a)` for a>=1. Therefore + +`||N-I||_infinity <= exp(-2a theta)+5 exp(-2a)`. + +This includes nonzero periodic diagonal aliases in the tail; no alias is silently omitted from this bound. + +For p>=101, theta>=1/(4p), and `a>=4 log(16p)`, + +`1-exp(-2a theta) >= 1-exp(-1/(2p)) >=1/(4p)`, + +`5exp(-2a)<=5/(16p)^8<1/(8p)`. + +Hence N has row diagonal-dominance margin at least `1/(8p)` in the following robust sense: after deleting its tiny nonzero diagonal aliases, its diagonal is 1 and its off-diagonal row sum is at most `1-1/(8p)`. In particular the fixed-width inverse theorem is not needed to supply the retained strict row dominance. + +Choose the legal integer alpha by + +`alpha=ceil(sqrt((4/pi)log(16p)))`. + +Then a has the required lower bound and is O(log p). The old sufficient condition `u>=ceil(log2(8p))` need not be reused: the direct estimate above replaces that prerequisite. + +## 2. Actual rounded H, interior matrices, and guard range + +Use exactly the retained structure: + +- `w=ceil(sqrt(16p/u))+2`, u=alpha^2=a/pi; +- cell weights `a_j=exp(-a(1/4-beta_j^2)/theta)`; +- one globally fixed downward dyadic Toeplitz kernel `Ghat_delta` for |delta|<=w, zero outside, with Ghat_0=1; +- positive downward dyadic weights ahat_j; +- within a physical cell, define H_jk **exactly** as `ahat_j Ghat_(j-k)/ahat_k`; +- across cells, use the retained downward Gaussian entries; set H_jj=1. + +Thus the actual interior block being inverted is + +`A_cell=diag(ahat) Ghat_cell diag(ahat)^-1`. + +It is not the unrounded Gaussian matrix, and its retained truncated Ghat is not silently replaced by the infinite Gaussian kernel. + +The weight range is + +`0<=-log a_j<=a/(4theta)=O(a d)` + +under theta between 1/(4d) and 1/(2d-1). The old eventual condition u*d<=p holds whenever d=p^epsilon, epsilon<1, and u=O(log p). It bounds the old rational perturbation calculation by the same `2^(O(p)) eta` allowance. The stronger range `O(a d)` should be used for grouped tensor precision, rather than charging O(p) independently for every axis. + +Choose the coefficient precision high enough that `||H-N||<1/(16p)`, including the Gaussian truncation, omitted diagonal aliases and weight-ratio rounding. The retained bound `p*2^(16p+7)*eta+2^(-46p)` at eta=2^(-32768p) is more than sufficient. Then H has diagonal 1, row gap at least `1/(16p)`, row norm below 2, and inverse norm at most 16p. All clipped principal windows inherit the same row gap. The physical period cut must remain a cell boundary, as in the retained bulk review. + +## 3. Schur complement bounds without an a*theta>1 assumption + +A scalar elimination step on a strictly row-diagonally-dominant matrix cannot decrease its minimum absolute row gap. For pivot k, + +`gap_i(new) >= gap_i(old)+(|a_ik|/|a_kk|) gap_k(old)`. + +The full absolute row sum also cannot increase, since the added update row has off-pivot sum less than |a_kk|. Induction applies regardless of the signs of filled entries. Since the diagonal begins real positive and the correction magnitude is below it, the surviving diagonal remains positive. + +Consequently the exact Schur complement + +`S=H_BB-H_BI H_II^-1 H_IB` + +has row gap at least `1/(16p)`, row norm below 2, positive diagonal, and + +`||S^-1||_infinity<=16p`. + +Principal interior inverses also have norm at most 16p. Rounding S by a row perturbation below 1/(32p) preserves gap at least 1/(32p) and inverse norm at most 32p. A row has O(w) nonzeros; the retained P-grid readily meets this target. These are polynomial bounds on arbitrary signed inputs, with no factor equal to the number of cells. + +The boundary geometry is unchanged: keep first/last w vertices of each physical cell, yielding O(s*w/d) boundary vertices and circular half-bandwidth at most 2w. Schur fill only joins the two end groups of one cell, while original cross-cell entries join neighboring end groups. Actual inverse and LU coefficient tables must be precomputed for this H/S, not imported from the old parameter values. + +Thus the retained banded/cyclic boundary solver's gap-dependent proof remains applicable with fixed constants enlarged. The proposed batched boundary arithmetic still needs its own data-layout and Strassen error audit, but there is no analytic conditioning obstruction at logarithmic a. + +## 4. Apply the shallow circuit directly to Ghat inverse + +Because a is eventually larger than 4pi, the symmetric Toeplitz matrix Ghat_cell satisfies + +`max(||Ghat_cell-I||_1,||Ghat_cell-I||_infinity)<1/8`. + +Downward rounding only decreases this off-diagonal sum. The shallow full-Toeplitz theorem therefore applies directly to Ghat_cell. Reverse the resulting invertible circuit and invert its gates to obtain Ghat_cell inverse. This avoids the two-branch Gohberg-Semencul difference entirely. The interior inverse is: + +1. multiply by diag(ahat)^-1; +2. apply the reversed shallow circuit for Ghat_cell; +3. multiply by diag(ahat). + +Its only superpolynomial individual scale comes from the **local** weight range O(a d). Completed A_cell inverse has polynomial norm by the strict row gap. Across g axes, the uncompleted local weight contribution is at most exp(O(g*a*d)); choosing `g<=p/(d polylog p)` with sufficient logarithmic slack keeps the total data precision O(p). The shallow circuit contributes polynomial-in-d amplification per axis, hence only O(g log d) additional logarithmic range. This is a guard calculation, not a complete simultaneous-layer routing proof. + +## 5. Independent shallow-Toeplitz algebra audit and a length caveat + +The parity-split triangular identity, block Schur displacement formulas, Stein-generator telescoping identity, and bounded-prefix induction in SHALLOW-TOEPLITZ.md check algebraically. In particular the parity children run in parallel, avoiding the earlier erroneous sequential-depth recurrence. The full block-LU recursion also runs its two children in parallel, with O(log N) displacement rank after O(log N) levels. + +That document currently states its circuit theorem for dyadic N. Actual cell interiors are not generally dyadic. Padding every tensor axis to its next power of two would be invalid. A plausible exact extension has been communicated to its author: split an arbitrary N into balanced sizes differing by at most one; a near-square rectangular Toeplitz dirty shear is a square Toeplitz shear plus one extra row/column rank-one shear. The latter can be implemented by a binary tree of unitary 2-coordinate reductions and its inverse, without clean ancilla. The rectangular Stein identity telescopes as before. This extension needs its own written proof/controls before it is used as an established interface. + +No new final kappa is asserted by this audit. + +## 6. Dyadic interior subdivision resolves the length caveat + +No arbitrary-length shallow FFT theorem is actually needed. Refine the interior/boundary partition inside each original physical phase cell of length L as follows. Keep its first and last w vertices as boundary. Starting with the remaining interval of length R=L-2w: + +- while R>4w, choose the largest power of two B<=R-w; +- mark the next B vertices as one interior, the following w vertices as boundary, and replace R by R-B-w; +- once R<=4w, mark all remaining vertices as boundary. + +Every interior now has exactly dyadic length. If R>4w, then `B>(R-w)/2`, so the next remainder is less than R/2. There are `O(1+log(L/w))` interiors and the total boundary count in the cell is `O(w(1+log(L/w)))`. The final boundary group, including the initially reserved last w vertices, has size at most 5w; all inserted separators have size w. + +The width-w matrix H cannot connect two different new interiors: their separating boundary gap has w vertices, so their nearest physical coordinates differ by w+1. Each new interior is still a principal block of the SAME globally fixed Toeplitz-similar cell matrix. Its weights retain the original cell gauge; there is no new rescaling or rerounding at a subdivision boundary. + +Eliminating one dyadic interior fills only between its immediately neighboring boundary groups. Each such group has O(w) vertices. Original H edges also connect only the same group or neighboring groups; an interior has length greater than w in the loop above. Thus, in inherited physical boundary order, the new Schur complement has half-bandwidth O(w), not O(w log d). Only its number of rows grows by O(log d): + +`n_boundary=O((s/d) w log d)`. + +Clipped endpoint cells of size O(w) may be declared wholly boundary, adding only O(w) rows; preserve the original physical period cut. This is the same endpoint convention as the retained bulk review. + +The boundary solve and coupling work therefore gain only polylogarithmic factors. All dyadic interior blocks use the established SHALLOW-TOEPLITZ theorem directly, without padding any tensor axis and without requiring a non-dyadic Fourier circuit. This resolves the length caveat in section 5 for the intended phase-cell application; it does not prove the separate simultaneous tensor-layer routing claim. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MANIFEST-UPDATE.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MANIFEST-UPDATE.json new file mode 100644 index 0000000000..423d53d1eb --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MANIFEST-UPDATE.json @@ -0,0 +1,172 @@ +{ + "ALGEBRAIC-STABILITY-INDEPENDENT.md": "7873e7d9a99fdc9de3ee224b9925d3e277e9773d7f5f8c2d0f1c19b0e52bc87f", + "BILATERAL-LDU.md": "0d65e66d227464dda1a19f781ae0a0be671cbfd347c41e37ec954f3d629d8a77", + "CAPPED-GAUGE-AUDIT.md": "1030a34922709ab8d3434a4c57a4a58934ae268c237fb598cb415697da411c9f", + "CAPPED-INTERIORS.md": "983c5cab2d2398f2192a8aec07daecb655f2de136e12a37f75672a14d27172ba", + "CAPPED-OUTER-ARITHMETIC.json": 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transfer at tau=19/8.", + "limitations": [ + "All finite supplier, common-frame, whole-call, semantic precision, fixed-tape and prime-packing assumptions of the first transfer remain.", + "No generated competitive finite matrix formula or practical runtime claim is made; the fixed formula search is effective but may have enormous constant cost.", + "This is a local conditional corollary, not an upstream selected certificate or an unconditional integer-multiplication theorem." + ], + "reviews": [ + { + "reviewer": "root agent", + "result": "Read matrix corollary and independent layout signoff; reproduced exact parameters/ledger and matrix construction controls; approved integrated conditional corollary." + }, + { + "reviewer": "matrix_boundary agent", + "result": "Read all 1022 supplied source lines and audited effective algebraic coefficient, fixed-tape, precision and rank-exponent transfer." + }, + { + "reviewer": "global_macro agent", + "result": "Independent new-parameter no-sort union, metadata, q-ordering and microhalo layout signoff." + }, + { + "reviewer": "phase_assembly agent", + "result": "Read both complete matrix transfer notes and independent layout signoff; constructed and replayed 47-slack ledger and six rejecting controls." + }, + { + "reviewer": "independent_fusion_review agent and independent children", + "result": "No blocking gap found under stated assumptions after full compiler, support, Gaussian, layout and tape-cost review. Generic SPD scale wording corrected without changing parameters. See independent FINAL-REVIEW.json." + } + ], + "arithmetic": { + "positive_slacks": 47, + "rejecting_controls": 6, + "minimum_saving": "0.0016733601154057784538170479517203762842018888767028492264912297200" + }, + "files": [ + { + "bundle_filename": "phase_fusion_note.tex", + "source_path": "work/phase_assembly_ultra/phase_fusion_note.tex", + "sha256": "08ecb1876695beee25484158e14cfd57c3d48f551a8a41d5abc998d71d674555", + "bytes": 33757 + }, + { + "bundle_filename": "MATRIX-COROLLARY.md", + "source_path": "work/phase_assembly_ultra/MATRIX-COROLLARY.md", + "sha256": "2d7fb1088e58bb86e574277733e082f135362215a69add71d9090213d32c9221", + "bytes": 5787 + }, + { + "bundle_filename": "check_matrix_corollary.py", + "source_path": "work/phase_assembly_ultra/check_matrix_corollary.py", + "sha256": "30db2d8e0f40fcc5c91e25f74cdc54e46aa4a0353c725345f9bcf5744e0b617f", + "bytes": 6692 + }, + { + "bundle_filename": "MATRIX-COROLLARY-ARITHMETIC.json", + "source_path": "work/phase_assembly_ultra/MATRIX-COROLLARY-ARITHMETIC.json", + "sha256": "c6efc4d2f6d17159cefebf5c2dd6099a6327fe0911c5dd87623cad96c7d6c56a", + "bytes": 16783 + }, + { + "bundle_filename": "PROOF-AND-COVERAGE.md", + "source_path": "work/matrix_boundary_ultra/PROOF-AND-COVERAGE.md", + "sha256": "db7509d754bb7fc3381419e0fed2674f22a9d57d34e8420fd5afa3c688fde9aa", + "bytes": 12130 + }, + { + "bundle_filename": "ALGEBRAIC-STABILITY-INDEPENDENT.md", + "source_path": "work/matrix_boundary_ultra/ALGEBRAIC-STABILITY-INDEPENDENT.md", + "sha256": "7873e7d9a99fdc9de3ee224b9925d3e277e9773d7f5f8c2d0f1c19b0e52bc87f", + "bytes": 11743 + }, + { + "bundle_filename": "MATRIX-SOURCE-COVERAGE.json", + "source_path": "work/matrix_boundary_ultra/SOURCE-COVERAGE.json", + "sha256": "fdc6849f97999a6cf99cdaa50591d24f78d6fb0bc580315f205452e5906b35cc", + "bytes": 2839 + }, + { + "bundle_filename": "MATRIX-SOURCE-CONSTRUCTIONS.json", + "source_path": "work/matrix_boundary_ultra/SOURCE-CONSTRUCTIONS.json", + "sha256": "114c1d8fb70592455a80c73d8a1bc6bfc9aca4e7395a420b6331d5474b4d1917", + "bytes": 21018 + }, + { + "bundle_filename": "MATRIX-TRANSFER-MANIFEST.json", + "source_path": "work/matrix_boundary_ultra/MANIFEST.json", + "sha256": "35d98d7579a16dae36530b4e5f4d0b8f925a069daa3f6fe1ea5276916167e0c2", + "bytes": 1159 + }, + { + "bundle_filename": "MATRIX-LAYOUT-SIGNOFF.md", + "source_path": "work/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md", + "sha256": "4c8c62f723232e842dc3bbdcc4defcc8ff1aeb86ba9f694661899dc6783b13c4", + "bytes": 3081 + }, + { + "bundle_filename": "MATRIX-LAYOUT-CONTROLS.json", + "source_path": "work/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json", + "sha256": "f8bd8729cb8438188e90fb9149f598b825461c97b4ed290584ca4367f812a48a", + "bytes": 1446 + }, + { + "bundle_filename": "PHASE-LAYOUT-AUDIT.md", + "source_path": "work/phase_layout_audit_ultra/AUDIT.md", + "sha256": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", + "bytes": 11774 + }, + { + "bundle_filename": "CAPPED-SIGNOFF.md", + "source_path": "work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md", + "sha256": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a", + "bytes": 4414 + }, + { + "bundle_filename": "CAPPED-GAUGE-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md", + "sha256": "1030a34922709ab8d3434a4c57a4a58934ae268c237fb598cb415697da411c9f", + "bytes": 9176 + }, + { + "bundle_filename": "LOG-A-PHASE-SCHUR-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md", + "sha256": "e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272", + "bytes": 10642 + }, + { + "bundle_filename": "FORWARD-SMALL-A-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md", + "sha256": "e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0", + "bytes": 6689 + }, + { + "bundle_filename": "SHALLOW-TOEPLITZ.md", + "source_path": "work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md", + "sha256": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310", + "bytes": 15480 + }, + { + "bundle_filename": "TENSOR-COMPILER.md", + "source_path": "work/inplace_algebra_ultra/TENSOR-COMPILER.md", + "sha256": "4d3b8466598a864a234b978e25976648d4e761840aad9153dd18d73d3dfe4d15", + "bytes": 13430 + }, + { + "bundle_filename": "SUPPLIER-SUPPORT-AUDIT.md", + "source_path": "work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md", + "sha256": "bd5b06eca043c5c871227264f6a954473c55cd16ec85bb420ff538fb8ec1d8d6", + "bytes": 16557 + }, + { + "bundle_filename": "SUPPORT-ROUTER.md", + "source_path": "work/halo_layout_ultra/SUPPORT-ROUTER.md", + "sha256": "6ddb322e11883c1177ba6cd1a77d23a1dac827d7290b32eaf642457b1d762a54", + "bytes": 7755 + } + ], + "post_review_correction": "Generic SPD prefix bound records both M and 1/m, not condition number alone; actual Gaussian bounds already satisfy it." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MATRIX-COROLLARY.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MATRIX-COROLLARY.md new file mode 100644 index 0000000000..f08bb57f9c --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MATRIX-COROLLARY.md @@ -0,0 +1,72 @@ +# Additional corollary using the supplied matrix-exponent theorem + +Status: **PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW**. The independent matrix-transfer and new-parameter layout audits passed, followed by the integrated root review and exact ledger replay. The first, Strassen-based result `kappa=0.001672922349588871` remains separately reviewed and unchanged. Neither result changes the upstream selected certificate. + +This corollary has an **additional hypothesis**: the supplied *Matrix Multiplication in Nine Fourths* theorem gives `omega_C<=9/4`. Its complete-source audit is `PROOF-AND-COVERAGE.md`; the effective approximate-bit transfer is `ALGEBRAIC-STABILITY-INDEPENDENT.md`. The result here retains every supplier and all-size assumption of the first phase-fusion theorem. + +## Effective matrix operation with the required error interface + +Fix `tau=19/8>9/4`. The matrix exponent theorem supplies a fixed block dimension m and rank r with `r0`. All local coefficient pages remain polynomial in p, while `L^g` remains superpolynomial. Microedge correction has exponent `epsilon+beta-3<-1.75`, including the generous old scalar arithmetic bound. Sparse microhalo key passes remain explicitly paid inside that negligible budget. + +All forward q-image group embeddings occur in native axis-major order before the one global low-bit gather. All compression q-image extractions occur after the inverse global gather. The entire Gaussian calculation keeps full dyadic target coordinate ranges, so every globally invoked H stage has complete coarse labels. This schedule needs no per-group gathering of the whole long-axis fields; the q-image scans have the already-paid Vd/g cost. + +The **optional full-group-key sort surcharge on the sparse union is omitted** because it is absent from the actual algorithm. The reviewed union routine extracts the complete rectangle `U_i=E_i x [L]^(g-1)`, transposes only its one polynomial-size field, computes its axis operator, and merges it into U in the original order. It does not radix-sort U. Reintroducing that sort is a rejecting control in the new ledger; it would destroy this parameter choice. This omission relies on the existing positive layout proof, not on the matrix theorem. + +`check_matrix_corollary.py` verifies 47 exact positive slacks and six rejecting controls and writes `MATRIX-COROLLARY-ARITHMETIC.json`. The rejected variants are endpoint equality, zero group guard, zero union slack, the old Vd row, the old full-cell data gauge, and adding a full-key sort to U. The first theorem's records are not overwritten. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MATRIX-LAYOUT-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MATRIX-LAYOUT-CONTROLS.json new file mode 100644 index 0000000000..aee3270196 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MATRIX-LAYOUT-CONTROLS.json @@ -0,0 +1,30 @@ +{ + "status": "PASS_MATRIX_COROLLARY_NO_SORT_LAYOUT", + "source_sha256": "b0d4648e16df74fc0a3644baa8a82cbb82842c0aa21abe915a28c0c4c949182a", + "exponents": { + "union_fraction": { + "rational": "-3430918384747396741111767383/8006701850307232414091700000000", + "decimal": "-0.000428505825356260251146382720850643577327042013532977151967005" + }, + "membership": { + "rational": "269791328968194081006854064986174254715733926/1069348903949477650676785089185287127357866963", + "decimal": "0.252294950667420906279768139101314143625086172719573255573065" + }, + "total_union": { + "rational": "320264641396534864583914893734048954038319544983689789/320804671184843295203035526755586138207360088900000000", + "decimal": "0.998316639884594221546182952048279623715798111123297150773509" + }, + "group_scans": { + "rational": "2402698356122892965636680000000000000000000000/3208046711848432952030355267555861382073600889", + "decimal": "0.748959903622628611922902526129555589081788674143596616501459" + }, + "microhalo_fraction": { + "rational": "-380335532915361620704585/80067018503072324140917", + "decimal": "-4.75021475791267813012557319136042532178866352100676648857598" + }, + "rejected_union_full_key": { + "rational": "3334408963974161672450196077539/2668900616769077471363900000000", + "decimal": "1.24935673626196560962328042591872403463400943697970053427205" + } + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MATRIX-LAYOUT-SIGNOFF.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MATRIX-LAYOUT-SIGNOFF.md new file mode 100644 index 0000000000..d683e4d495 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/MATRIX-LAYOUT-SIGNOFF.md @@ -0,0 +1,64 @@ +# Matrix-corollary layout: positive signoff + +The no-sort union layout in `AUDIT.md` applies unchanged to +`work/phase_assembly_ultra/MATRIX-COROLLARY.md`. Its algebraic conditions are +closed axis supports E_i and a complete microbox; neither its proof nor its +implementation requires gkappa`. The minimum excess is greater than `10^-11`. Consequently every fixed logarithmic factor and each use of the unconditional quasi-linear short arithmetic kernel can be absorbed in a single positive fraction of that excess. Floors, rounded alpha, and dyadic L alter constants or logarithmic factors. All claims are eventual in p; one fixed finite prefix of inputs can use the retained elementary fallback. This statement becomes an admitted theorem only when the named implementation hypotheses receive the intended full review. + +## Parameter convention + +Let `a_bit=0.001733427039589844234749` and `b=0.001675462676808103522925` be the retained lower bounds certified by the selected PR1094 replay. For applying the older compact recurrence, lower the usable bit saving to `a_eff=(b+q)/2`; this keeps its historical convention `a_eff0` with `(1-t_stop)b>a_eff` and `q<(1-t_stop)b`. + +For each visited coarse block and group, all required local coefficient pages can instead be regenerated in polynomial time in p: there are g local windows of size at most L=Theta(p^8), their rational origins have O(p/d) bits, and the published local coefficient/Schur/circuit constructors are polynomial in that size and requested precision. Recover local phases and anchor gauges from these origins; do not form a raw uncentered global q² exponential on the data. The page contains O(p)-bit online coefficients. Its polynomial setup and sequential loading are amortized by the complete `L^g` payload, which eventually dominates every fixed power of p. This gives an explicit access method even when the coarse labels are permuted; no random global coefficient-table lookup is assumed. + +Global chirps, twists, positive diagonal normalizations and exact scale factors are separable products. The prefix-product scan supplies their products in O(V polylog p), with completed scalar range `O(d alpha^2)=o(p)`. Packed polynomial products retain their saving epsilon. The finite supplier, exact common-label frame, whole-call hosting, semantic depth bound with C1=1, phase precision, prime packing and fixed-tape input/output assumptions are inherited unchanged. They are still conditional interfaces, as in the selected result. + +## Cost ledger, with conservative wide bandwidth + +Let `h_w=1/2` and `h_K=beta+1-epsilon`. Normalize all costs by V. Fixed logarithmic factors are absorbed by strict gaps. + +| Component | power of p | reason | +|---|---:|---| +| main transform | `1-epsilon*q` | deferred reservations | +| constant global coordinate routing | `1-a_eff` | paid support/full router | +| dense Gaussian groups | `epsilon-q*beta` | `(d/g) g^(1-q)` | +| grouped bit routing | `epsilon-a_eff*beta` | same grouping, bit supplier | +| one/fixed group scans | `epsilon-beta` | d/g groups | +| boundary algebra | `(omega-1)*h_w` | batched w-column Strassen | +| sparse union, basic movement | `beta+h_w` | V g w overall | +| sparse union, conservative key surcharge | `beta+h_w+h_K` | full active-group keys | +| polynomial products | `1-epsilon` | retained short polynomial products | +| core-face corrections | `epsilon+beta+3-8+2` | old local maps on sparse copies, with a loose p² allowance | + +Use the rational upper bound `omega<45/16`, checked by `7^16<2^45`. The target balances the first and third rows exactly before a fixed 10^-11 absorption backoff. + +This ledger is accepted only after its circuit tensorizer, grouped sparse route, target-grid core/halo adapter, forward small-a inverse bound, and deferred-reservation interfaces have written proofs. 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b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md new file mode 100644 index 0000000000..bbeeee7431 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md @@ -0,0 +1,242 @@ +# Grouped phase-cell layout audit + +The sparse-union layout in `REFINED-PHASE-FUSION.md` has a constructive +implementation with its stated `V g^2 W polylog(p)/d` per-group movement +budget. It does not require sorting the full tensor or retaining d full +payload scans. The conclusion is conditional on the separately supplied +dense local tensor compiler, rounded phase-cell factorization, and precision +bounds. This audit does not prove those algebraic/analytic inputs. + +The maximum physical bandwidth **W**, including the broad forward map, must +be used in the dyadic carving and support sets below. The narrower inverse +bandwidth alone is insufficient for the broad map. The phase-assembly +branch confirms this convention. + +## Exact operator order and the union that is actually invariant + +Write H in interior/boundary order as `[A B; C D]` and let +`S=D-C A^-1 B`. The displayed identity in the parent note is correct. In +execution order its inverse can be scheduled as + +``` +diag(A^-1,I) +[I 0; -C I] +diag(I,S^-1) +diag(A,I) +[I -B; 0 I] +diag(A^-1,I). +``` + +The middle two diagonal factors commute. Thus there are three dense +interior stages and two sparse windows: the C and S^-1 stages share one +window; B uses the second window. For H itself, reverse and invert this +sequence: the same original B,C and S occur, with signs reversed and dense +A/A^-1 interchanged. No dense A^-1 B or C A^-1 shear is smuggled into the +sparse boundary cost. + +For each active axis i, let J_i be its designated boundary coordinates. +Let E_i contain J_i and every interior coordinate incident to an original +nonzero entry of B_i or C_i. Original physical bandwidth W implies that E_i +lies within O(W) of the carved endpoints. Include physical-period cuts, +artificial phase cuts and clipped microbox endpoints. Dyadic carving gives + +``` +|E_i| <= C (L/d+1) W log d, +mu := max_i |E_i|/L = O(W log d/d + W log d/L) +``` + +inside a length-L microbox. All sparse factors have the block form +`T_i = I_(E_i^c) direct_sum T_i|E_i`. In particular S_i^-1 can be dense on +J_i: J_i is already included in E_i. Let + +``` +U = {x in [L]^g : some x_i belongs to E_i}. +``` + +Then `|U|/L^g <= g mu`. Every sparse tensor-axis operation preserves U and +its complement. If x_i is changed by a nonzero off-diagonal entry, both +its old and new coordinates belong to E_i; otherwise it is unchanged. +This proves closure for an arbitrarily long sequence of these sparse +operations, not just one application. + +Dense A or A^-1 is **not** covered. It can move a near-boundary interior +coordinate into the deep interior. Scatter U before a dense stage and +regather afterwards. `check_union.py` includes an exact example in which +dense A^-1 sends union-supported data outside U. The actual schedule above +therefore uses two union gathers, not one retained union for all six stages. + +## Complete rectangles inside the union: no missing-zero assumption + +For one sparse axis i define + +``` +U_i = {x : x_i in E_i} = E_i x [L]^(g-1). +``` + +This is a **complete rectangle**, and it is wholly present in U regardless +of which other coordinates are near a boundary. All nonidentity data needed +by T_i is in U_i. A line with every other coordinate outside its own E_j +still has every E_i coordinate present. A line with another boundary +coordinate may contain more data, but those extra E_i^c entries are identity +coordinates and need no boundary computation. + +Keep a copy of U in its original microbox lexicographic order, with a short +coordinate/membership header. For each axis i: + +1. Scan U once and copy records with x_i in E_i to an auxiliary stream. + The copied stream is lexicographic E_i x [L]^(g-1). +2. Rename the increasing E_i coordinates by their ordinal 0,...,|E_i|-1. + This is a stream counter; no rank/select query is needed. +3. Expose that one field at the line suffix using the retained small-field + transpose. Its length is at most L=poly(p), so only O(log p) binary + block moves are needed. Pad only this field to its next power of two + temporarily, at a factor below two, and remove that padding before the + next axis. The other fields may be arbitrary spectators. +4. Apply the original B/C or boundary S/S^-1 factor on its complete E_i + line, retaining identity coordinates E_i\J_i when appropriate. +5. Undo the transpose at the actual field length. Merge the ordered E_i + stream back into a fresh copy of U, replacing exactly those entries. + +No array is completed by invented zero coordinates. The finite checker +verifies the extracted rectangle cardinality and the full tensor result. + +The extraction/merge costs O(|U|) per axis. Transposition costs +O(|U_i| polylog p). Consequently one sparse window costs + +``` +O(V [g^2 mu + g mu polylog p]) + = O(V g^2 W polylog p/d) + O(V g^2 W polylog p/L). +``` + +There is no factor `K=g log s` from a computed-key sort in this part of the +algorithm. A more general radix implementation would still fit with extra +small powers of g, but the complete-rectangle construction avoids it. + +## Union extraction and piece catalogues + +Only one full microbox scan is needed to form U, and one to scatter it. +For every complete p-bit coefficient record, compute its g membership flags +from its coordinate tuple and OR them. The flags and coordinate header have +O(g log L) bits. A nonzero number of flags sends the whole record once to U; +it is not duplicated for each active axis. + +This is a charged descriptor calculation. For actual source periods of +bitlength h=Theta(p/d), q_j inversion and phase-cell endpoint calculations +use O(h^2 polylog p) elementary bit operations per axis. Within a phase +cell, the greedy dyadic carve is found by at most O(log d) halvings and +integer comparisons. Thus a conservative membership cost is +`O(g h^2 polylog p)` per record, which is o(p) for +`g=p^(1-epsilon-r-zeta)` and epsilon>2/3. At the proposed epsilon this has +power about 0.005. It is absorbed by reading the p-bit record once. +Alternatively the original remainder phase generator and an odometer give +the same membership flags in sequential coordinate order. + +Factor pages may be regenerated once per complete microbox/group from its +actual coarse origins, using the supplied polynomial-time coefficient +generation. The complete inner microbox and untouched spectator payload +are superpolynomial in p and amortize this polynomial setup. This avoids +random access to a global catalogue in the scrambled coarse order created +by a support-limited gather. Online row/factor pages are then visited in +the ordered extracted E_i line. All page rewinds are within that line or +RHS batch and are included in the corresponding boundary arithmetic. + +## Batched right-hand sides do not add a large transpose + +All other inner coordinates and following tensor axes are independent +right-hand sides for an axis operation. Their count R is superpolynomial. +Batch them in a dyadic width between W and 2W; the final incomplete batch +can be padded once at constant cost. To put the batch index outside the +boundary-row field, move only that polynomial-width row field. This costs +O(log p) times the boundary volume, not a transpose of the long spectator +field. Each W-by-W factor page is scanned once per RHS batch and reused in +the block product. Reverse the small-field layout after the solve. + +Thus the assumed Strassen `W^omega` per W RHS gives the quoted total +`O(V W^(omega-1) polylog p)` boundary arithmetic. This audit checks the +layout and factor reuse; stability and the no-pivot factor theorem remain +the phase-assembly branch's responsibility. + +## q_j embeddings, selections and source order need one group scan + +For a group, each coordinate map `j -> q_j=floor(tj/s+1/2)` is increasing +and injective. Its tensor product preserves lexicographic order on its +image. Enumerate the target-group rectangle once. At target coordinate k, +set + +``` +j=max(0,ceil(s(2k-1)/(2t))). +``` + +The coordinate belongs to the image iff `j9/4`. + +If the surrounding proof allows polylog p factors, the bandwidth's polylogs remain polylogs; no positive power of p is introduced by coefficient approximation or error amplification. The algorithm provides absolute accuracy, which is what local residual bounds use. It makes no relative-error promise for nearly canceled output entries. + +The recursion can also be scheduled with sequential tapes. At one fixed-base node, its fixed number of input block combinations and output block sums require a fixed number of complete scans and fixed block permutations of `O(n^2)` records. Store child inputs consecutively, process children depth-first, and stream their returned blocks into accumulators. Copying a pending-child region or repositioning a finite set of tapes costs another `O(n^2)`-record pass at that node, already covered by the recurrence's addition term. Addresses and recursion descriptions add at most logarithmic factors. If any implementation retains an additional `polylog W` factor, choose the rank scheme first with exponent strictly between 9/4 and the stated target tau, leaving enough fixed exponent slack to absorb it. No algebraic unit-cost random-access assumption is needed for the target power tau. + +## 4. Boundary substitution and the outer ledger + +The existing capped boundary algebra uses W simultaneous right-hand sides. A square product costs `O(W^tau M(p))`; dividing by W² processed entries gives `O(W^(tau-2) M(p))` per entry. One axis has `O(V W/B)` boundary entries, and there are d axes. Its total is consequently + +`O(V (d/B) W^(tau-1) [M(p)/p] polylog p)` + +when V is the total bit volume, as in the phase-assembly notes. There are V/Theta(p) scalar entries, so the factor M(p)/p is the per-bit arithmetic charge. Equivalently, if V counts scalar entries instead, the formula has M(p) without the division. The retained quasi-linear short multiplier has M(p)/p=polylog p. + +This retains the essential d/B factor. With `d=p^epsilon`, `B=p^b`, and `W=p^(1/2) polylog p`, its normalized exponent is + +`epsilon-b+(tau-1)/2`. + +Taking `tau=19/8` gives `(tau-1)/2=11/16` and boundary saving `5/16+b-epsilon`. This is already more than enough to put the basic sparse-union cost near the binding row at b near 3/4. Pushing tau down to the unattained endpoint 9/4 is unnecessary. + +The changed matrix algorithm has the same input/output interface as the former Strassen product. It does not change the original Gaussian matrix, boundary support, anchored gauge, square-block inverse choice, or number of right-hand sides. The surrounding local-residual proof must still supply polynomial bounds on local exact inverses and partial right-hand sides. The new product's error is `poly(p)2^-P`, so that proof applies with a fixed larger guard constant. + +The optional full-coordinate-key surcharge in the old arithmetic ledger would become binding if mechanically retained at b near 3/4. It is not part of the actual rectangle extraction/one-field transpose/ordered-merge algorithm, as confirmed by the separate phase-layout audit. Dropping it is a distinct layout fact, not a consequence of matrix multiplication. It must remain explicitly justified in the final outer ledger. All retained supplier assumptions remain in force. + +No final improved outer exponent is claimed in this note; that requires the whole ledger and the independent group-layout/gauge checks. + +## 5. Exact controls run in this audit + +`check_source_constructions.py` uses only Python's exact rational arithmetic and writes `SOURCE-CONSTRUCTIONS.json`: + +* Finite Fourier/separation support for M=1,...,12: the modular relation has no wraparound and zero weight is exactly g=h,u=v. +* Determinant change of basis for a=1,...,8 and b=2,...,8, for every first-input monomial: quotient and kernel diagonal slices are exactly the two claimed neighboring convolution tensors; the upper-right block vanishes. +* Three-sector identity for a,h=1,...,14: all weights are nonnegative and each of the three retained branches is exactly the claimed tensor after its permitted identification. +* Exact rational diagonal product lower bound through a=300. + +All controls passed. They detect several relevant implementation mistakes but are not a finite substitute for the universal arguments above or for the theorem's nonconstructive rank-existence step. + +## 6. Deliberate limitations + +There is no generated competitive finite matrix scheme, no numeric claim about its constants, no use of a rank formula at exponent exactly 9/4, and no fast algorithm for finding the first successful algebraic tuple. The algorithm remains computable and uniform because the finite search halts under the proved existence statement and its cost is independent of the growing input. The outer conclusion remains conditional on exactly the previously retained supplier/computational assumptions. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/PUBLICATION-CHECKS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/PUBLICATION-CHECKS.json new file mode 100644 index 0000000000..befac93552 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/PUBLICATION-CHECKS.json @@ -0,0 +1,226 @@ +{ + "status": "PASS_PORTABLE_EXACT_PHASE_FUSION_PACKAGE", + "scope": "Exact finite controls and arithmetic plus retained native receipt binding. This does not replay the entire native dependency or discharge all-size hypotheses. The matrix corollary retains an additional external premise.", + "package_files": 180, + "headline_kappa": "1677160151714381/1000000000000000000", + "checks": [ + { + "script": "frontier_fusion/check_frontier_fusion.py", + "artifact": "frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "frontier_fusion/independent_substitution_check.py", + "artifact": "frontier_fusion/INDEPENDENT-SUBSTITUTION-RECEIPT.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/inplace_algebra_ultra/check_exact_algebra.py", + "artifact": "evidence/inplace_algebra_ultra/EXACT-ALGEBRA.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/inplace_algebra_ultra/check_tensor_compiler.py", + "artifact": "evidence/inplace_algebra_ultra/TENSOR-COMPILER-CONTROLS.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/inplace_algebra_ultra/check_supplier_capacity.py", + "artifact": 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"36f8c00c200cc375512ad0e881e91b42daba10c962fce37465040cb717b5cc60" + } + }, + "environment": { + "python": "3.14.7", + "platform": "macOS-15.7.7-arm64-arm-64bit-Mach-O", + "dependencies": { + "numpy": "2.5.4", + "scipy": "1.18.1", + "sympy": "1.14.0" + } + }, + "native_replay_mode": "Fresh native providers and retained parent; no reuse/cache flags. Complete source verification was run before publication.", + "capsule_generator_replay": "Regenerated canonical capsule byte-identical to committed capsule using explicit --native-source and --native-work paths.", + "registered_python_312": { + "python": "3.12.15", + "sympy": "1.14.0", + "status": "PASS_REGISTERED_RESEARCH_PACKAGE", + "command": "python3.12 scripts/research_package.py --package research/capped-phase-fusion", + "exact_control_replays": 21 + }, + "native_build_requirements": "C++17 compiler and Boost headers; successful run used /opt/homebrew/include for header lookup." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/PUBLICATION-SOURCES.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/PUBLICATION-SOURCES.json new file mode 100644 index 0000000000..3f760bee8a --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/PUBLICATION-SOURCES.json @@ -0,0 +1,12 @@ +{ + "research_base": "e6f9716621f4e01fc190617934cdfaf072bd976f", + "publication_base": "a5d4f08ac102d1a6da301d89379c75285eca680e", + "supplier": { + "repository": "https://github.com/CrocSwap/integer-mult-bounds", + "pr": 1661, + "commit": "f852af5e178ce035cd46181a29d9330bcbcaa206", + "directory": "research/gmx-t74-p566-ep400", + "manifest_sha256": "d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f" + }, + "scope": "Source snapshots and exact recipe for complete cold replay; historical native receipt hashes are external provenance, not all bundled artifacts." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/README.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/README.md new file mode 100644 index 0000000000..f7f3348ee3 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/README.md @@ -0,0 +1,162 @@ +# Capped phase-cell fusion: conditional final kappa = 0.001677160151714381 + +This package gives the conditional final integer-multiplication bound + +\[ +T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad +\kappa=\frac{1677160151714381}{10^{18}}=0.001677160151714381. +\] + +The contribution changes the **outer Gaussian resampling reduction**. It uses +shallow in-place Toeplitz circuits, a common Hadamard schedule for multiple +axes, recentered and capped Gaussian cells, and explicitly charged sparse +boundary operations. The finite supplier is the unchanged, pinned +[PR1661](https://github.com/CrocSwap/integer-mult-bounds/pull/1661) T74 construction. +Its complete native verification was run afresh, including both new provider +words and the retained parent. The final exponent is a conditional theorem +from the written transfer argument; finite checks alone do not prove it. + +Read [the proof](phase_fusion_note.tex), especially its T74 substitution section, +and the [independent substitution review](frontier_fusion/INDEPENDENT-SUBSTITUTION-REVIEW.md). +The proof's earlier numerical specializations remain valid historical cases. +“Independent” in this package means separate agent implementations/reviews, +not an independent human referee or a proof-assistant verification. + +## Exact paid inputs and result + +| Quantity | Value | +|---|---:| +| Retained bit saving c | 0.001733427039589844234749 | +| Chosen complex saving b | 0.00167971337478 | +| Effective tensor saving q | b - 10^-10 | +| Effective bit saving a | (q+b)/2 | +| Native dimension m | 73 | +| Native live width W | 66304 | +| Final kappa | 0.001677160151714381 | + +The complete native profile is +`{1:1559766, 2:337668, 3:278058, 4:84912, 5:54612, 8:5328, 48:22748}`. +All decimals above are exact rationals. Exact epsilon, xi, beta, every outer +cost power and all 45 strict inequalities are in +[FRONTIER-FUSION-ARITHMETIC.json](frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json). +Five deliberate bad variants reject. The native moment has an outward rational +upper bound below one; a larger test endpoint is rejected. + +For the same pinned supplier, PR1661's original outer reduction gives +0.0016768966690212993077507..., so the new outer transfer improves its final +exponent by about 2.6348269308e-7. This is an exponent comparison, not a measured +runtime improvement or a claim of global priority. The selected-result record +is not changed by this contribution. + +The supplementary branch gives 0.001677600129305161 **only with the additional +assumption omega_C <= 9/4**, used through a fixed algebraic algorithm of exponent +19/8. That extra premise is not needed for the headline Strassen-based result. +The supplied 9/4 paper, its audit and historical local locator are recorded in +PROOF-AND-COVERAGE.md and MATRIX-SOURCE-COVERAGE.json; it is treated as an explicit +additional premise, not as a theorem certified by this package. + +## Scope and remaining assumptions + +Retained assumptions include the all-size compiler, exact common-label and +mutable-source realization, whole-call hosting, the semantic precision guard +with C1=1, fixed-tape/prime-packing interfaces, paid arbitrary-width equal-chunk +interchange, exact recovery, and the original analytic transfer. The construction +does not prove an unconditional multiplication theorem, a complete Lean theorem, +or a practical finite cutoff. The written proofs address the new outer steps; +finite algebra and layout checks support, but do not replace, those arguments. + +Original assignment: improve the bound using broad mathematical resources. +For repository indexing this is a changed scope of `uniform-transfer-audit`: +it constructs and audits a replacement outer transfer rather than only checking +one inherited implication. [TASK.json](TASK.json) records the scope and next steps. + +## Reproduce + +Python 3.12, with SymPy 1.14.0, is the registered environment. The exact checks +also ran locally with the Python version recorded in PUBLICATION-CHECKS.json. +The package verifier checks its manifest and source pins, works in a temporary +copy, and compares the regenerated mathematical receipts. It does not modify +this directory. From the repository root: + +```sh +python3 -m pip install sympy==1.14.0 +python3 -B research/capped-phase-fusion/verify.py +python3 scripts/research_package.py --package research/capped-phase-fusion --check-only +``` + +The first command after installation is the bounded outer-proof evidence replay, +not a cold native-supplier replay. The native run already completed with all 12 +top-level stages passing, 319680 owners, prefix 1441 charged at 2048, physical row +coefficient 19545 below 20161, and rejected controls: 114 admission, 12 bit, +15 controller, 8 independent owner, 6 geometry. The retained compact RESULT +and [native binding](frontier_fusion/NATIVE-ADMISSION-BINDING.json) record it. + +To reproduce the full native dependency, obtain the exact source commit +`f852af5e178ce035cd46181a29d9330bcbcaa206` from CrocSwap/integer-mult-bounds, +install its documented dependencies (NumPy, SciPy, SymPy, a C++17 compiler and +Boost headers), and run its unmodified command without cache/reuse flags: + +```sh +python3 -B /absolute/pr1661/research/gmx-t74-p566-ep400/verify.py \ + --work /absolute/new-native-work --audit-shards 4 +python3 -B research/capped-phase-fusion/verify.py \ + --native-source /absolute/pr1661/research/gmx-t74-p566-ep400 \ + --native-work /absolute/new-native-work +``` + +The second command binds a completed external native run to this package's +source and mathematical result. It compares regenerated proof-relevant content; +fresh timings and log bytes need not match the historical run. The full 121 MB +assembly and vendor tree are not duplicated here. The receipt capsule adds canonical hashes for 46 JSON receipts. The 58 hashes +in the historical native binding identify external run artifacts; they are not all bundled files. +The original bind_native_admission.py is retained as historical provenance; +use the new top-level verifier for portable binding. + +PACKAGE-MANIFEST.json is the authoritative publication manifest. MANIFEST.json, +MANIFEST-UPDATE.json and earlier review receipts are explicitly historical +snapshots. The copied upstream CLAIM.json is immutable and contains older pending +language; the later completed native RESULT records this contribution's replay. +Hosted CI and independent human review are separate from the local checks. + +## Sources and overlap + +Prepared by **Vlado Vojdanovski with substantial AI assistance**. New material is +Apache-2.0. The original manuscript and source-specific notices retain their +licenses. This is not an official OpenAI release or endorsement. + +The main research snapshot was `e6f9716621f4e01fc190617934cdfaf072bd976f`; +the publication base is recorded in SOURCES.json. Exact T74 source pins are in +PUBLICATION-SOURCES.json and the source manifest. +The [unaltered T74 NOTICE](target002_upstream/pr1661/research/gmx-t74-p566-ep400/NOTICE) +is included. The original +OpenAI #109 framework and the Harvey–van der Hoeven reduction remain inputs. + +The in-place algebra builds on Giorgi–Grenet–Roche, +[*Fast In-place Algorithms for Polynomial Operations*](https://arxiv.org/abs/2002.10304), +and Dumas–Grenet, [*Fast in-place accumulation*](https://arxiv.org/abs/2302.13600). +SHALLOW-TOEPLITZ.md distinguishes their results from the new parity-split, +low-displacement and prefix-stability arguments. The Gaussian factorization +also relates to +[PR489](https://github.com/CrocSwap/integer-mult-bounds/pull/489), which already +gives a branch-free local Gaussian inverse but leaves joint evaluation, +routing and boundary composition open. The present package supplies a different +in-place tensor compiler and capped boundary transfer for those outer costs. +[PR1141](https://github.com/CrocSwap/integer-mult-bounds/pull/1141) concerns a +reversible quadratic phase stream and is related background, not a dependency. +The overlap check does not establish novelty beyond these identified sources. + +## Open next questions + +1. Can a new finite supplier exceed the present complex rate by a substantial + amount? Even a lossless outer transfer cannot reach 0.002 from b<0.00168. + First step: emit a complete side circuit for a lower-cost decoder and measure + its actual rank profile, helper storage and restoration. The missing piece is + the physical circuit, not an idealized central-rank estimate. +2. Can the capped Gaussian transfer be independently checked or formalized with + its complete fixed-tape schedule? First step: independently reconstruct the + sparse-union and support-local routing lemmas. Existing finite tests cover + algebraic identities, not an end-to-end all-size tape implementation. +3. Can the same outer transfer consume a future reviewed supplier without + changing its proof? First step: rebind the endpoint, prefix and completed-call + restoration contract, then replay the exact ledger. The supplier must satisfy + the full functional interface; a numerical endpoint alone is insufficient. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/REVIEW-RECEIPT.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/REVIEW-RECEIPT.json new file mode 100644 index 0000000000..2521d02ada --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/REVIEW-RECEIPT.json @@ -0,0 +1,53 @@ +{ + "status": "PASS_CONDITIONAL_PHASE_FUSION_REVIEW", + "created_utc": "2026-10-14T02:06:22.293042+00:00", + "kappa": "1672922349588871/1000000000000000000", + "kappa_decimal": "0.001672922349588871", + "upstream_selected_unchanged": true, + "scope": "Local all-size conditional transfer, retaining the selected finite suppliers and common-frame, whole-call, precision, fixed-tape and prime-packing hypotheses.", + "reviews": [ + { + "reviewer": "root agent", + "result": "Read standalone TeX, integrated notes and exact ledger; cap, supplier and layout audits passed; authorized local conditional status." + }, + { + "reviewer": "gaussian_fast agent", + "result": "Independent capped-gauge and finite-block audit passed; exact inherited-weight reanchoring and broad derivatives checked." + }, + { + "reviewer": "global_macro agent", + "result": "Independent capped carving, sparse-union, q-image and microedge layout audit passed." + }, + { + "reviewer": "inplace_algebra agent", + "result": "Global selected-H support and capacity audit passed; all g controls need not fit one original axis." + }, + { + "reviewer": "phase_assembly agent", + "result": "Read named proofs and relevant retained source contracts; reconstructed complete cost ledger and reviewed capped interfaces." + } + ], + "phase_assembly_replays": [ + "check_capped_outer.py: 44 positive exact slacks and 5 rejecting controls", + "check_forward_cell.py: 350 cases, 5184 factorization and 16496 completion entries", + "check_union.py: 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"cb9e1da38400b658f9209e465b4e5896fadd17dc66afa862f38bae8d8c1f2058" + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/ROOT-EXACT-REPLAY.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/ROOT-EXACT-REPLAY.json new file mode 100644 index 0000000000..4f80ff6ecc --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/ROOT-EXACT-REPLAY.json @@ -0,0 +1,39 @@ +{ + "status": "PASS_ROOT_EXACT_BASELINE_AND_CAPPED_MARGIN_REPLAY", + "selected_commit": "e6f9716621f4e01fc190617934cdfaf072bd976f", + "baseline_complex_moment_upper_minus_one": "-2785787033843904352780990891572049605608726093667665717994302747547920313/2135987035920910082395021706169552114602704522356652769947041607822219725780640550022962086936576", + "q_below_seven_step_supplier": 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Mathematical transfer depends on the accompanying proved circuit/layout/precision lemmas and inherited repository assumptions." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/ROOT-MATRIX-EXACT-REPLAY.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/ROOT-MATRIX-EXACT-REPLAY.json new file mode 100644 index 0000000000..bd509b8377 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/ROOT-MATRIX-EXACT-REPLAY.json @@ -0,0 +1,21 @@ +{ + "status": "PASS_ROOT_MATRIX_CONDITIONAL_LEDGER", + "additional_assumption": "supplied complex matrix multiplication exponent <=9/4, used with strict 19/8 algorithm exponent", + "kappa": "836680052702889/500000000000000000", + "gain_over_selected": "34995416443/50000000000000000", + "epsilon": "3204021080669390695403468340153200000000000000/3208046711848432952030355267555861382073600889", + "cap": "6006737803538363603786961767383/8006701850307232414091700000000", + "group": "19999559600000000000000/80067018503072324140917", + "slacks": { + "main": "16040234287175308991291365140577147466485957516731679/1604023355924216476015177633777930691036800444500000000000000000", + "Gaussian": "16040234287175308991291365140577147466485957516731679/1604023355924216476015177633777930691036800444500000000000000000", + "global routing": "84100856093020917/40000000000000000000000", + "grouped routing": "36073302400837752235461073644407147466485957516731679/1604023355924216476015177633777930691036800444500000000000000000", + "scans": "399990069170827316553169929437873914639085240389379902516731679/1604023355924216476015177633777930691036800444500000000000000000", + "matrix boundary": "33307674429019250694161274836955713292695810709060319172243893/534674451974738825338392544592643563678933481500000000000000000", + "union": "16040249599476451935460767629144447487515470930957516731679/1604023355924216476015177633777930691036800444500000000000000000", + "pointwise products": "1599326431642752671058066465736543223602284795889379902516731679/1604023355924216476015177633777930691036800444500000000000000000", + "microfaces": "1470245804132326338277304830206155329249828857963126634172243893/534674451974738825338392544592643563678933481500000000000000000" + }, + "sparse_fraction_slack": "3430918384747396741111767383/8006701850307232414091700000000" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SHALLOW-TOEPLITZ.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SHALLOW-TOEPLITZ.md new file mode 100644 index 0000000000..bc4349b2da --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SHALLOW-TOEPLITZ.md @@ -0,0 +1,197 @@ +# Shallow, stable, dimension-preserving Toeplitz circuits + +Status: a positive algebraic circuit construction for **dyadic local tiles**. It gives the missing full-Toeplitz primitive in the normalized Sylvester merge. It does not reconcile the anchor coordinates with the original Gaussian gauges; that remains a separate obligation. The construction uses no data ancilla, no zero-initialized coordinate and no fan-out. Fixed coefficient generation is allowed polynomial time in the local tile size and precision. + +## 1. Result + +Let N be a power of two, and let H be an N by N real or complex Toeplitz matrix with + + max(||H-I||_1, ||H-I||_infinity) <= delta < 1/4. + +There is an N-wire linear circuit applying H, with O(log^4 N) rounds of disjoint invertible 2 by 2 gates and diagonal gates, O(N log^4 N) gates, and polynomial-in-N bounds for all coefficients and all partial circuit operator norms and inverse operator norms. Constants depend only on delta. + +For any requested output accuracy 2^-p, rounding the coefficients to p+O(log N) bits gives operator error at most 2^-p; arithmetic at that precision also gives uniform forward error 2^-p on bounded inputs, after allowing the same O(log N) guard bits. A rational approximation to the input Toeplitz coefficients and all required gate coefficients can be generated using poly(N,p) bit operations. This coefficient preprocessing may use polynomial extra memory; the **data circuit** still has exactly N live coordinates. + +The same result applies to upper or lower triangular Toeplitz matrices with max(||T-I||_1,||T-I||_infinity)<=eta for **any fixed eta<1**, with only O(log^2 N) rounds and constants depending on eta. In particular eta=1/2 is permitted for the generator-normalization step below, even though the full-Toeplitz statement uses delta<1/4. Descendants need bounded norm and inverse, not continued distance below 1/4. This independent triangular construction is important: a contiguous bottom-recursion / shear / top-recursion has linear depth even though its work is quasi-linear. + +## 2. Dirty Toeplitz shear is a shallow circuit + +For disjoint m-coordinate vectors x and y, implement + + (x,y) -> (x,y+Tx), + +where T is Toeplitz, in O(log m) rounds when m is dyadic. It is an invertible square map on the existing 2m wires. + +Write the first column and row of T as c_0,...,c_(m-1) and c_0,r_1,...,r_(m-1). Set + + g_0=h_0=c_0/2, + g_j=(c_j+r_(m-j))/2, + h_j=(c_j-r_(m-j))/2, 1 <= j < m. + +Then T=C(g)+C_-(h), where C is a circulant and C_- is a negacyclic circulant. For a unitary m-point FFT F, C(g)=F^* diag(DFT(g)) F. A circulant shear therefore consists of F on x and y in parallel, the disjoint pairwise shears y_j+=DFT(g)_j x_j, and F^* on x and y. Every FFT stage consists of disjoint unitary 2 by 2 butterflies; bit reversal is a disjoint swap stage. + +For gamma=exp(pi i/m), D=diag(1,gamma,...,gamma^(m-1)), + + C_-(h)=D^-1 C(Dh) D. + +Use diagonal twists before and after the same circulant-shear circuit. The two completed shears commute. Source x is restored after each. No circulant inverse is needed, so zero Fourier multipliers cause no problem. Every gate is invertible even when T itself is singular. + +All intermediate norms are polynomial in m and the l1 norm of the Toeplitz symbol. In the 2-norm, FFT and twist stages are unitary, and the spectral diagonal has norm at most the symbol l1 norm. + +## 3. Independent triangular primitive: parity splitting + +Let T_N(f) be lower triangular Toeplitz multiplication by f modulo z^N. Write + + f(z)=e(z^2)+z o(z^2). + +Reorder the existing coordinates into even and odd powers, N=2m. Then + + T_N(f) = [ E ZO ; O E ], + +where E=T_m(e), O=T_m(o), and Z=T_m(z). All these m by m triangular Toeplitz matrices commute. Set + + k=o/e mod z^m, + s=e-z o^2/e mod z^m. + +The exact factorization is + + [ E ZO ; O E ] + = [ I 0 ; T_m(k) I ] + diag(T_m(e),T_m(s)) + [ I T_m(zk) ; 0 I ]. + +Thus apply the right Toeplitz shear, the two child triangular circuits **in parallel**, and the left Toeplitz shear. The recurrence for rounds is + + D_tri(N) <= D_tri(N/2)+O(log N), + +giving D_tri(N)=O(log^2 N). The two recursions act on disjoint coordinates. Inverses are obtained by reversing the gate sequence and inverting every gate, at the same depth. + +Stability is not proved by multiplying coefficient l1 bounds afresh at each depth. Each child is a principal submatrix or principal Schur complement of the original triangular matrix, after a coordinate permutation. If the initial T_N(f) differs from I in both 1- and infinity-norm by at most eta<1, every descendant M satisfies + + ||M-I|| <= eta+eta^2/(1-eta)=eta/(1-eta) + +in both norms, and ||M^-1||_2 <= 1/(1-eta). This follows directly by expressing the descendant as a Schur complement of a principal block of the original matrix; nested elimination and restriction preserve that description. Hence all local diagonal blocks and local Schur multipliers have constant 2-norms. Their triangular Toeplitz coefficient l1 norms are at most sqrt(N) times these constant bounds. The dirty shear circuits from section 2 therefore remain polynomially bounded. + +## 4. Full Toeplitz recursion: displacement rank grows additively + +Use the Sylvester shift displacement + + Delta_m(M)=Z_m M-M Z_m. + +A Toeplitz H has rank Delta_N(H)<=2. At a node of size 2m, partition + + M=[ A B ; C D ], + Z_(2m)=[ Z 0 ; E Z ], E=e_0 e_(m-1)^T, + K=A^-1 B, L=C A^-1, S=D-C A^-1 B. + +The usual block factorization is + + M=[ I 0 ; L I ] diag(A,S) [ I K ; 0 I ]. + +Again A and S run in parallel. To bound ranks, let U=[-C A^-1,I] and V=[-A^-1 B;I]. Then M V=[0;S], U M=[0,S], and direct block multiplication gives + + Delta_m(S)=U Delta_(2m)(M) V-S E A^-1 B. + +Therefore rank Delta(S)<=rank Delta(M)+1. Also, the upper-left block of Delta_(2m)(M) is Delta_m(A)-B E, so rank Delta(A)<=rank Delta(M)+1. At depth d, both children thus have displacement rank at most 2+d. + +The top-right block of Delta(M) equals Delta(B), while its bottom-left equals E A+Delta(C)-D E. Consequently + + rank Delta(B)<=r, + rank Delta(C)<=r+2, + rank Delta(K)<=rank Delta(A)+rank Delta(B)<=2r+1, + rank Delta(L)<=rank Delta(C)+rank Delta(A)<=2r+3. + +All shear matrices have displacement rank O(log N). + +Every recursive M is a principal Schur complement of the original H. Since H=I+R and both ||R||_1 and ||R||_infinity are at most delta, + + max(||M-I||_1,||M-I||_infinity) <= delta/(1-delta), + ||M^-1||_2 <= 1/(1-delta). + +The inverse assertion also follows because M^-1 is a principal submatrix of the inverse of a principal submatrix of H. Thus A,S,K,L are uniformly bounded in 2-norm independently of recursion depth. + +## 5. Bounded-generator shear compiler + +For an m by m matrix T define its Stein displacement + + E_T=T-Z T Z^T=-Delta_m(T) Z^T+T e_0 e_0^T. + +Its rank is at most rank Delta(T)+1. Choose an orthonormal basis g_1,...,g_r for the column space of E_T, and put h_j=E_T^* g_j. With conjugates interpreted appropriately, + + E_T=sum_j g_j h_j^*, + T=sum_j L(g_j) U(conj(h_j)). + +Here L(g) is lower triangular Toeplitz with first column g, and U(h) is upper triangular Toeplitz with first row h. The second identity is the finite telescoping sum T=sum_(k=0)^(m-1) Z^k E_T (Z^T)^k. All g_j have 2-norm 1, and all h_j have 2-norm at most ||E_T||_2=O(1). Their coefficient l1 norms are O(sqrt(m)). + +A zero or tiny first coefficient of h is harmless. Choose an integer gamma >= 2||h||_1 (a fixed conservative gamma=64N suffices under the stated bounds), and set + + P=I+U(h)/gamma. + +P and P^-1 have constant norm, and P is upper triangular Toeplitz. On the same two existing vectors, one generator term is implemented by + + x <- P x, + y <- y+gamma L(g) x, + x <- P^-1 x, + y <- y-gamma L(g) x. + +The result is y<-y+L(g)U(h)x and x is restored. Source transforms use section 3; the other two operations use section 2. Each term costs O(log^2 m) rounds. There are O(log N) terms, hence O(log^3 N) rounds for a full low-displacement-rank shear. + +Although the partial generator sum can be larger than its final T, its norm is at most the sum of O(log N) products of O(sqrt N)-norm factors, hence O(N log N). Within a term the source norm remains constant times its entry norm and the target norm is polynomially bounded. Terms are completed as shears with one common source, so their partial sums add; one must not charge a product of polynomial factors for every generator. + +The full recurrence is therefore + + D_full(N) <= D_full(N/2)+O(log^3 N)=O(log^4 N). + +## 6. Global prefixes, rounding and preprocessing + +At each block-LU node the completed right shear, completed middle block, and completed left shear have constant norm and constant inverse norm. At an intermediate time there is at most one unfinished generator compiler or one unfinished triangular compiler along each recursive path; disjoint paths form a direct sum. If B(N) bounds a full circuit prefix and inverse prefix, this gives B(N)<=max(C B(N/2),N^C), hence B(N)=N^O(1). The same argument applies to suffixes and to the triangular compiler. + +There are N polylog(N) arithmetic gates. Perturbation of one bounded gate by epsilon contributes at most N^C epsilon to the whole operator, by the prefix/suffix bounds. The sum over all gates remains N^O(1) epsilon. Therefore p+O(log N) coefficient/arithmetic bits suffice; there is no N or N^2 precision term on the data. + +The coefficients are fixed before seeing the data. One concrete polynomial-time generation route is: + +1. Approximate the original Toeplitz symbol by dyadic rationals at p+O(log N) bits, preserving Toeplitz structure. The operator perturbation is at most N times the entry perturbation. +2. Compute every recursive Schur matrix as a principal Schur complement of this original rational matrix. Its entries are ratios of minors of that original matrix, so numerator and denominator bitlengths are polynomial in N and p. Avoid an unnecessarily expanded recursive rational expression. +3. For each Stein displacement, use exact rational elimination to select independent columns and rational, unnormalized Gram-Schmidt to obtain orthogonal vectors v_j. Then g_j=v_j/sqrt(v_j^*v_j) and h_j=E_T^*v_j/sqrt(v_j^*v_j). The resulting generators have the bounded norms proved above. Nonzero rational residual norms are at least 2^-poly(N,p), so poly(N,p) offline precision suffices for their normalization. Equivalently a high-accuracy SVD may be used. +4. Compute the finite series divisions and FFT roots to p+O(log N) final bits. All final data-gate coefficients are polynomially bounded. Offline intermediate rational numerators may be longer; they never become data wires. + +This separates polynomial-time coefficient setup from bounded-precision data application. Applying an algebraic-RAM theorem directly to arbitrarily large integers would not provide this separation. + +## 7. Normalized Sylvester merge + +For two b-point children, N=2b, the normalized map supplied by the radical-reduction branch is + + S(B,A)=f(z)B(z)+z^b g(z^-1)A(z), + f(z)=product_lower(1+(x/zeta)z), + g(z^-1)=product_upper(1+(zeta/x)z^-1). + +Let h=g(z^-1)/f(z), K_ij=h_(i-b-j), H_ij=h_(i-j), 0<=i,j0. Every principal submatrix has spectrum in [mu,M]. Every principal Schur complement S also does: S^-1 is a principal block of the inverse of a principal submatrix, whose spectrum is in [1/M,1/mu]. Thus every recursive A,S has norm at most M and inverse norm at most 1/mu. The shears K,L have norm at most kappa=M/mu. The displacement-rank formulas are unchanged. + +Use orthonormal g generators and h of norm O(kappa), and choose gamma>=2||h||_1. The triangular source transform P is still near I, so the depth O(log^4 N) is unchanged. Generator-stage amplitudes and coefficients are poly(N,kappa). Across the O(log N) completed ancestor factors, the bound becomes + + log(max prefix/inverse-prefix norm) + = O(log N * log(2+M+1/mu) + log N). + +If M and 1/mu are exp(O(u)), this is O((u+1)log N). The corresponding data precision is p+O((u+1)log N), and g-axis tensorization adds O(g(u+1)log N) guard bits. This is a usable broad-Gaussian extension once a separate spectral argument supplies the stated eigenvalue interval. Reversing the circuit applies H^-1; the inverse itself need not be Toeplitz. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SOURCES.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SOURCES.json new file mode 100644 index 0000000000..a5d007aae1 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SOURCES.json @@ -0,0 +1,18 @@ +{ + "schema_version": 1, + "base_commit": "a5d4f08ac102d1a6da301d89379c75285eca680e", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11", + "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1", + "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075" + } + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md new file mode 100644 index 0000000000..72404c42d7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md @@ -0,0 +1,139 @@ +# Independent audit: support-local simultaneous-H supplier + +Conclusion: the retained compact phase supplier can be used at width g with cost + + O(V g^(1-q) polylog(p)) + +without the previously charged additive V p^(1-a), **provided the call is made globally on the complete dyadic tensor, not separately on one fixed coarse-label microbox**. This is conditional on the same accepted finite complex circuit and arbitrary-width equal-chunk interchange as the retained supplier, and on the support-router transfer proved in `work/halo_layout_ultra/SUPPORT-ROUTER.md`. The new support-router's unequal-spacing algebra and its use here have been checked below. + +The full address still has O(p) bits. That quantity remains in descriptor arithmetic, keys and error precision; it no longer determines the width of any newly required fast coordinate interchange. + +## 1. What the actual phase frames touch + +The primary source is `work/integer-mult-current/upstream/build/sections/05-layers.tex`. + +At lines 86–139, a residual basis is a matrix M in GL(m,F_2), extended **inside F_2^m**. At width e=mf it is applied separately to the column + + (x_(1,j),...,x_(m,j)), 0<=j T as prescribed, + T\S -> S\T bijectively, + identity outside S union T + +is a coordinate permutation supported on at most 2gK positions. Its inverse restores the original coarse-coordinate interpretation. The target can be chosen before the payload and a reserved untouched suffix. No stable reordering of every coarse field is required. + +Apply the H supplier to this full global stream, whose complete coarse-coordinate ranges remain present. A fixed coarse-label microbox alone does **not** contain enough complete K-bit fields: its high coarse bits are fixed. Borrowing those fixed labels inside such a microbox would violate the supplier's complete-field hypothesis. This audit approves the global call only. + +The tensor compiler's H schedule is independent of all coarse labels. Those labels affect only diagonal coefficients between complete H calls. The diagonal scan can still be organized by coarse-label microboxes after undoing the global exposure permutation. Coarse-dependent coefficients therefore do not require separate supplier invocations. + +## 3. Deferred reservoirs and the row prefix + +At a phase with h unprocessed selected bits, reserve whole K-bit chunks for two front hG fields, one back hG field, and a leading row prefix. With K=128G, the field reservation is at most 3h/128+O(1) chunks. Row preparation needs O(log h) further chunks, including the product stock for nested bit and complex calls. For h above a fixed polylog(p) threshold the total R(h) is at most h/16. The remaining bounded/polylogarithmic h can be processed individually. + +Every such chunk is one of the current selected chunks. The phase applies C only to the other h-R(h) selected bits and identity to the reserved chunks; the latter are deferred to the next phase. + +This requires one explicit modification to the original exposition: **do not first apply C to the reserved row/front/back selected bits**. That preliminary work in the old all-axis construction only ensured those selected bits were eventually covered. It is not needed to create rows or complete temporary fields. Source `compact-control-layout.tex` lines 30–62 proves that the active invocation is the required C tensor separately on every row and identity on all reserved/spectator coordinates, for arbitrary incoming payloads. Hence dirty, unprocessed reserved selected bits are valid. + +The leading q0 complete chunks have cardinality 2^(q0 K). Choose q0 from the actual product row stock Q_rows=W_complex^D_complex W_bit^D_bit. This preserves the accepted nesting contract; replacing the product by a maximum would be incorrect. Pad the leading row count to a multiple of Q_rows once for this phase, by adding whole rows. The factor is below two. All descendants inherit the same complete within-row fields and the same row stock; no new independent bit coordinates are allocated. + +A completed active invocation is identity on the row label and every temporary coordinate. Each padded zero row is therefore zero at completion and can be deleted. Delete padding and restore the exposure layout **before the next deferred phase**. Consequently row padding does not multiply by 2 at every phase or tensor axis. + +The next deferred phase gathers only its h'=R(h) remaining neighborhoods. A gather/restore can again be extended with support at most 2h'K; it may move previously processed coordinates temporarily but fixes them after the inverse placement. Their values need not be recomputed. There is no need to move all p coordinates back into a stable global axis order. + +## 4. The logarithmic K choice is legal after tightening the statement + +The original compact-control proposition states the sufficient asymptotic condition K/log p -> infinity. K=128G does not satisfy that literal condition. The source proof, however, uses it only at lines 149–154 to obtain + + K >= G+4 ceil(log2 p)+10, + delta_bad <= 5/(128p^3). + +K=128G satisfies the displayed finite inequality directly. The earlier/later-source guard proof, complete-field layout, row stock, scalar gates and semantic guard have no other use of K/log p -> infinity. Thus the transfer must cite the tightened finite guard hypothesis; it must not claim that the new K satisfies the old stronger asymptotic wording. + +With this change, all compact interchanges retain width O(h log p), and the per-node movement overhead is O(V_node h^(1-a) polylog(p)). Here h is the current selected width, not the ambient number d of original Gaussian axes. The supplier's stopping threshold and branching analysis likewise use its own top width g; global p remains only in logarithmic guards and O(p)-bit descriptors. + +## 5. Check of the support-router extension + +The original arbitrary-source router colors active targets by their current location modulo K. Targets in one color have disjoint K-bit guards. The new router stores only the r active control digits, H=rG, even when the target spacings are unequal multiples of K. + +The four-update proof depends solely on disjointness of the guard intervals: on a good address the first y update adds 2zt within a guard, the t update adds its old low bit e without carry, and the second y update adds z(1-2(t+e)). Their net is z(1-2e), changing exactly the selected low bit. The final t update restores t. Nothing uses equality between successive target gaps. A complete F therefore fixes every nontarget source bit, even inside another guard. F; load u from the sources; F; unload implements the desired simultaneous XOR for sources outside the target set. + +All rotations remain globally bijective, because their controls precede the target after the prescribed back-field shuttle. Their current-control inverse is used on bad addresses. The ideal selected-bit XOR preserves the guard/control bad predicate, so the old sparse repair applies unchanged. Its O(p)-bit keys and polynomial per-record arithmetic remain explicit; they are not replaced by a unit-cost sort. + +For a permutation moving k positions, factor it into two involutions and three XOR stages per matching. Three disjoint 3kG-bit reservoirs can be chosen when n>=9kG; the three matching cohorts avoid the respective reservoirs. Each target residue class then uses width at most kG in each control field, rather than ceil(n/K)G. Original equal-width interchanges place and restore the fields. This gives O(V k^(1-a) polylog p), plus descriptor setup. + +When records are short, choose only ell_s=ceil(log2 p)^2 untouched binary bits, not a whole long original axis. Here is explicit noncircular suffix bookkeeping. For a requested permutation pi with moved support at most k, if n>2(k+1)ell_s, a gap in its support contains a consecutive ell_s-bit field S disjoint from the actual last ell_s bits. Use one original equal-ell_s interchange A to move S to the actual suffix. Route the conjugate A pi A^-1 on the n-ell_s prefix only, then undo A. The conjugate fixes that suffix pointwise because pi fixed S. Choose all control reservoirs inside the prefix, after this interchange, and recompute positions and residue classes after their placements. Thus no control operation touches the grouped suffix. Grouping its 2^ell_s complete values into a record preserves the data volume and gives a superpolynomial record length. If n<=2(k+1)ell_s, the retained full-span bound is already O(V(k+1)^(1-a) polylog p), so no suffix grouping is needed for this support bound. The placement costs O(V polylog p), not V ell^(1-a) for the original long axis length ell. + +For the calls here, k=O(hK)=O(h log p), and n/(k log p) tends to infinity because g=p^beta with beta<1 while n is of order p in the retained full tensor. If this inequality fails in a small regime, n=O(k polylog p), so the retained full-span bound is already O(V k^(1-a) polylog p). The support theorem's own temporary reservoirs may have O(kG) bits; this is a polylogarithmic enlargement of support, already priced in the same bound. + +## 6. Complete charge table + +| Operation | Active coordinate/data support | Charge | +| --- | --- | --- | +| Initial and inverse neighborhood exposure | At most 2gK moved bit positions | O(V g^(1-a) polylog p) | +| Deferred phase exposure at width h | At most 2hK positions | O(V h^(1-a) polylog p) | +| Fixed binary residual adapter | h selected bits; compact controls O(hG) | O(V_node h^(1-a) polylog p) | +| Control-field shuttle | At most O(hG)-bit chunks | Original O(V_node(hG)^(1-a)) | +| Modular rotations and exceptional repair | Full scan; O(p)-bit descriptors/keys | O(V_node) after record grouping | +| Residual child concatenation | Existing contiguous first-r slots | Descriptor change only | +| Source/sink/inverse phases | Selected-bit masks and coefficients | O(V_node) | +| Scalar role gates, parking and restoration | Fixed role streams at aligned addresses | O(V_node) | +| Whole-row padding and deletion | One current-phase row prefix | O(V), factor below two, removed before next phase | + +No row in this table requires an interchange of width p. + +Let tau=1-a. The retained stopped branching argument gives O(V h^(1-q) polylog p) for each phase, with q 2(k+1)L. +``` + +Indeed, after excluding the actual suffix, the at most k moved positions +cut the remaining interval into at most k+1 gaps. If every gap had length +less than L, that prefix would have length at most (k+1)L-1. The displayed +inequality contradicts this bound. + +Let A be the old disjoint equal-L swap exchanging S with the actual suffix. +Execute A, route the conjugated permutation A pi A^{-1}, then execute +A^{-1}. Because pi fixes S pointwise, its conjugate fixes the actual suffix +pointwise. It still moves at most k coordinates. All endpoints are now in +the n-L prefix. Choose the three dirty reservoirs in that prefix and run +all compact rotations, original control shuttles, elementary CNOTs, and +bad-hole repair there. Recompute endpoint names after A and after the +reservoir placement. In the application, n-L>=9kG holds with a wide +asymptotic margin. + +The suffix is a complete binary field with arbitrary contents. Reinterpreting +its 2^L values as part of a record gives record length R0*2^L, without +creating, padding, or deleting data. This length exceeds every fixed +polynomial in p. No operation whose cost relies on that record length acts +on its component coordinates. Both instances of A use the already approved +primitive and cost O(V L^(1-a))=O(V polylog p); they do not invoke the new +router. This removes the prospective circularity. + +For the general router outside this sparse branch, the retained +first-L/second-L/actual-last-L candidate convention and matching cohorts +also work. Arbitrary candidates should not be substituted blindly: a +candidate can partially overlap the actual suffix, whereas the old swap +requires equal disjoint fields. After each candidate placement, the same +n-L prefix exclusion is mandatory. If the entire binary span is only +polylogarithmic, the old elementary construction suffices. If +n=O((k+1)polylog p), the old full-span construction gives the desired +support bound up to polylogarithms; its own short-record preparation must +still use the old suffix construction. + +For a full H supplier call, reserve the suffix outside its selected field +and keep it fixed throughout the completed call. If necessary, perform +the same old L-bit placement before exposure, update the logical names of +the neighborhoods, and restore it after inverse exposure. The source +neighborhoods need not remain physically consecutive before exposure: +the exposure permutation explicitly routes their named bits. The target +complete chunks are placed before the reserved suffix. + +## 3. Whole-row padding and the next diagonal scan + +The leading row prefix and the front/back controls are assembled from +complete K-bit neighborhoods in the globally complete binary tensor. +Their bits are not assumed zero or semantically independent of coefficient +values. Completeness means every coordinate value is represented in the +stream. Multiplication by previous coarse-dependent diagonal coefficients +does not remove any address or change this property. + +Deferring the selected bits lying in reserved chunks is valid: the accepted +active invocation is the desired tensor on the other selected bits and +identity on every reserved coordinate, for arbitrary incoming coefficients. +The preliminary individual transforms in the old layout were used to +finish those selected bits, not to create complete row or control ranges. +After deferral, those transforms are performed in later phases instead. + +The row count is rounded once to a multiple of the actual product + +``` +Q_rows = W_complex^D_complex * W_bit^D_bit. +``` + +The padded row label is allowed to be a nonbinary preceding fiber count. +Every within-row field remains a full binary field. The supplier's +interchanges and controls act within those complete fields; the global +exposure router is not applied to the padded row label. Nested bit adapters +consume and restore the bit stock before later complex splits, exactly as +in the retained product-stock argument. + +At the completed invocation boundary, the operator is identity on the row +label. A padded zero row therefore returns to zero even though temporary +role computations may have combined or moved its contents. Delete those +rows before inverse exposure or the next deferred phase. This bounds each +phase's volume by twice its original volume, not by a factor exponential in +the number of phases. + +All H calls in the compiler have the same selected-bit schedule for every +coarse label; label dependence resides in the intervening diagonal tables. +Run one H call on the complete global tensor, complete every deferred +phase, remove padding, and restore the original exposed-coordinate order. +Only then run the next diagonal scan. The microbox layout used for that +scan is the restored layout in which the local coordinates are already +contiguous (or the separately paid low-bit gather layout). No padded row, +partial H result, or temporarily relocated coarse label is presented to +the diagonal-table lookup. Thus scanning different coarse-label tables +does not require separate H invocations or borrowing fixed labels inside +a microbox. + +## 4. The actual parameter inequalities + +The largest exposure moves k=O(gK)=O(p^.093 log p) binary coordinates. +With G=Theta(log p), its three dirty reservoirs require + +``` +9kG = O(p^.093 log^2 p) = o(p). +``` + +The suffix occupies only L=Theta(log^2 p) positions. In particular, + +``` +n/(kG) = Omega(p^.907/log^2 p) -> infinity, +n/((k+1)L) = Omega(p^.907/log^3 p) -> infinity. +``` + +There is ample room both for the old spectator placement and for all +subsequent prefix-only reservoirs. These inequalities concern the full +binary address, not any one original axis. + +The only per-axis inequality is + +``` +K + O(log p) < ell_axis = Theta(p^.002), +``` + +which holds eventually. The original K/log p -> infinity wording must be +replaced, as already noted in `SUPPLIER-SUPPORT-AUDIT.md`, by the finite +guard inequality K>=G+4ceil(log2 p)+10. K=128G satisfies it. + +The supplier's front/back controls are made by concatenating reserved +complete neighborhoods. Their chunk count is + +``` +ceil(2hG/K) + ceil(hG/K) <= 3h/128 + 2. +``` + +The leading row stock adds O(log h) complete chunks. Above a fixed +polylogarithmic cutoff this is at most h/16, and smaller h is handled +individually. No single original axis stores hG control bits. Indeed +gG/ell_axis=Theta(p^.091 log p) diverges; the construction works precisely +because it does not demand this false per-axis fit. + +Consequently the reviewed mechanisms support the proposed +O(V g^(1-q) polylog p) global-call cost under the retained supplier +assumptions. The suffix exclusion and completed-call boundary should be +kept explicit in any final assembly. They are necessary bookkeeping, +not a new exponent restriction or an unproved sorting primitive. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SUPPORT-ROUTER.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SUPPORT-ROUTER.md new file mode 100644 index 0000000000..4f39fa9eaf --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/SUPPORT-ROUTER.md @@ -0,0 +1,157 @@ +# Coordinate routing measured by moved support + +This is a conditional extension of the retained compact arbitrary-source +router. It uses the same equal-width chunk interchange with cost +`O(V H^(1-a))`, original fixed-tape modular rotations, counters and sparse +radix repair. It changes the packed-control length from the full address span +to the number of active target positions. No computed-key sort of the full +input is assumed. + +Let a supplied permutation of `n=O(p)` binary address coordinates move at +most `k` positions. The stream is a complete binary rectangle, with arbitrary +preceding fibers and following payload. In the regime used below, +`n/(k log p) -> infinity`; short remaining cases can use the retained +full-span router, since `n=O(k polylog p)` there. The conditional cost is + +``` +O(V (k+1)^(1-a) polylog p + poly(p)). +``` + +All other coordinate positions return to their original values. This is a +support bound, not a bound on how far a moved coordinate travels. + +## Unequally spaced target lemma + +Put `B=2^G`, `G=4 ceil(log2 p)+6`, and `K=64G`. Take target positions +`j_0<...2(k+1)ell`, a gap outside the moved support contains a contiguous ell-bit +field S disjoint from the actual last ell bits. Use an original equal-ell +interchange A to move S to that actual suffix. Route the conjugate +`A pi A^-1` on the n-ell prefix only, then undo A. The conjugate fixes the +suffix pointwise because pi fixed S. Choose every dirty control reservoir +inside the prefix after this interchange, then recompute positions and +residue colors. Thus no control operation touches the enlarged record. +Its 2^ell complete values absorb the polynomial descriptor work; no data +is padded or duplicated. If `n<=2(k+1)ell`, the retained full-span bound +already gives `O(V (k+1)^(1-a) polylog p)`. The placement itself costs only +O(V polylog p). This explicit suffix-conjugation bookkeeping was +independently supplied in `work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md`. + +## The gather uses only S union T + +Let S list the k selected low-coordinate bits in the desired tensor order, +and T list k consecutive destination slots. Send S bijectively to T. Send +the remaining domain slots `T\S` bijectively to the remaining output slots +`S\T`, and fix everything outside `S union T`. This explicitly extends the +requested gather to a coordinate permutation supported on at most 2k bits. + +The other coarse-coordinate bits can be scrambled by this extension. Their +order is retained and described by the inverse permutation; **no stable +compaction of all coarse fields is performed**. A coarse neighbor address +can be decoded, incremented in its logical axis, then encoded in this +physical coarse order when constructing a sparse correction key. + +## Evidence and boundary + +`check_sparse_shear.py` checks all 32,768 addresses for targets 0 and 6 with +guard length 5, including 512 good addresses, inverse composition and every +bad repair. It also checks 4,000 arbitrary/constructed-good addresses with +unequal gaps 9 and 11 and sources inside guard fields. `check_halo.py` +checks three explicit support-at-most-2k gather permutations. + +These controls support the algebra; the cost argument above is the +all-size conditional transfer. It retains the original bit-supplier, +equal-width interchange, fixed-tape and common-label hypotheses. It does +not upgrade an arbitrary full computed-key sort to a fast router. + +Sources: the retained reports `compact-arbitrary-source-routing.md` and +`review-arbitrary-routing.md` under +`work/integer-mult-current/references/semantic-bulk/rad20/reports/`. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/TASK.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/TASK.json new file mode 100644 index 0000000000..8fc25b910b --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/TASK.json @@ -0,0 +1,72 @@ +{ + "schema_version": 1, + "task_id": "uniform-transfer-audit", + "priority_revision": 3, + "base_commit": "a5d4f08ac102d1a6da301d89379c75285eca680e", + "original_card": { + "id": "uniform-transfer-audit", + "question": "Can one exact all-size/compiler, precision, hosting or restoration implication be proved or falsified?", + "starting_points": [ + "certificates/selected-result.json", + "docs/audit.md" + ], + "methods": [ + "Prove one missing implication with explicit quantified hypotheses.", + "Find an exact counterexample at the claimed interface boundary." + ], + "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.", + "first_step": { + "command": [ + "python3", + "-B", + "scripts/research_baseline.py" + ], + "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY", + "inspect": "Read the selected review scope and choose one exact all-size or precision implication.", + "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay." + }, + "first_experiment": "Write its quantified inputs and conclusion; find a missing hypothesis or construct a minimal exact adversarial instance.", + "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.", + "allowed_variations": [ + "Choose one tape, precision, hosting or restoration lemma." + ], + "scope_change_triggers": [ + "Replacing a quantified obligation with another finite smoke test." + ], + "overlap_terms": [ + "uniform transfer", + "all-size", + "precision" + ] + }, + "scope_status": "changed", + "actual_scope": "Construct and audit capped phase-cell fusion and substitute the pinned fully replayed T74 supplier to obtain final conditional kappa=0.001677160151714381.", + "scope_change_reason": "The user explicitly requested stronger bounds using broad resources; the resulting work replaces several outer transfer steps rather than auditing a single inherited implication.", + "bottleneck_link": "The outer reduction repeatedly scans Gaussian axes; in-place tensor compilation and capped sparse boundaries reduce that cost. The unchanged complex supplier still caps the possible final saving.", + "budget_minutes": 180, + "stop_rule": "Publish the verified conditional transfer with exact source bindings and bounded replay; retain larger-supplier research as open work.", + "outcome": "completed", + "evidence": "phase_fusion_note.tex; frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json; frontier_fusion/NATIVE-ADMISSION-BINDING.json; PUBLICATION-CHECKS.json", + "author": "Vlado Vojdanovski", + "ai_assistance": "Substantial AI assistance. No independent human or proof-assistant review claimed.", + "open_questions": [ + { + "question": "Can a new finite supplier materially exceed the present complex rate?", + "why_it_matters": "The current complex endpoint is below .00168 and cannot yield final .002.", + "first_step": "Emit and price a full native side circuit for a lower-cost central decoder.", + "blocked_by": "Actual routing, helper and restoration costs." + }, + { + "question": "Can the full capped transfer be independently formalized?", + "why_it_matters": "Finite algebra checks do not establish all-size tape complexity.", + "first_step": "Reconstruct the sparse-union and support-local routing proofs.", + "blocked_by": "Formal all-size schedule and inherited compiler interfaces." + }, + { + "question": "Can future suppliers be substituted without changing the outer proof?", + "why_it_matters": "The outer gain can compose with later finite improvements.", + "first_step": "Bind endpoint, prefix and completed-call restoration contracts and rerun the ledger.", + "blocked_by": "A complete supplier satisfying the same functional contract." + } + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/TENSOR-COMPILER.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/TENSOR-COMPILER.md new file mode 100644 index 0000000000..19efb9cb1c --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/TENSOR-COMPILER.md @@ -0,0 +1,160 @@ +# Tensorizing local circuits without dimension padding + +Status: an algebraic compiler and scan-cost lemma. It reduces a tensor product of different bounded-depth local circuits to a bounded sequence of simultaneous binary Hadamard calls and diagonal scans on the **same** data volume. The retained supplier must support the stated Hadamard call on selected local bits and restore the agreed endpoint record order. No new fast-transform supplier is asserted here. + +## 1. Interface and conclusion + +Let N=2^m and let g axes each have N local coordinates. A coarse-label block contains V=N^g data records in lexicographic local-coordinate order. On axis a let C_a be an N-wire linear circuit of depth D; every layer consists of disjoint invertible two-coordinate gates and one-coordinate diagonals. The permitted gates are bounded shears, swaps, FFT butterflies, and diagonals. Circuits, pair matchings, coefficients and depths can differ by axis and by coarse label. Their tables are fixed independently of data values. + +Write H=[[1,1],[1,-1]]. A simultaneous Hadamard call means the tensor product of one H acting on one selected bit in each axis, with all remaining local and coarse coordinates as spectators. It acts on existing 2^g-coordinate fibers inside the N^g records; it allocates no new coordinate. The bit selected in an axis may be prescribed by the compiler. + +After padding shorter circuits by identity **operations**, the tensor product C_1 tensor ... tensor C_g is a sequence of + + at most (8m+2)D simultaneous H calls + +and O(Dm) tensor-diagonal scans. A tensor diagonal has the form + + d(i_1,...,i_g;c)=product_(a=1)^g d_a(i_a;c), + +where c is the fixed coarse label. Each diagonal scan uses O(1) amortized coefficient multiplications and O(1) amortized coefficient-table record movements per data record, after loading gN local values. The number of arithmetic-data coordinates remains exactly N^g throughout. + +If all local gate coefficients and inverses are bounded by B and all original prefix and inverse-prefix norms by Lambda, the compiler's local prefix bounds are Lambda times poly(B), up to absolute constants. For polynomial B and Lambda this is N^O(1). The tensor-prefix logarithmic bound is O(g log N). More generally depth D=polylog(p), B=poly(p), N=poly(p) gives an O(g polylog(p)) guard even without a stronger prefix theorem. + +An outer similarity whose logarithmic range is Q_a on axis a adds only sum_a Q_a. Therefore the retained phase-cell case Q_a=O(a d) uses + + p + O(g a d + g polylog(p)) + +working bits. The condition g a d+g polylog(p)=O(p) is sufficient for O(p)-bit records. + +## 2. Fixed Hadamard words, including zero shears + +All formulas below are matrix-product order; execution reads right to left. Let U(t)=[[1,t],[0,1]]. Direct multiplication gives + + H diag(2,1) H diag(-3,1) H = [[-8,-10],[0,-6]], + diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H = U(5/4). + +For q nonzero, conjugate by A_q=diag(4q/5,1): + + U(q)=A_q diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H A_q^-1. + +This has three H factors. To include t=0 with uniformly invertible diagonals, choose a known B>=|t| and c=2B+1. Then + + U(t)=U(t+c) U(-c). + +Both parameters have magnitude at least B+1 and at most 3B+1. This is a **six-H word for every t**, with all scalar coefficients and inverses bounded by poly(B). There is no division by t and no case split at t=0. Transpose the word to obtain the same six-H count for lower shears. + +There are two bounded identity words: + + I = (1/2) H H, + I = diag((1-i)/4,(1+i)/4) H diag(1,i) H diag(1,i) H. + +The second identity uses three H factors. Its middle product before the leading diagonal equals diag(2+2i,2-2i). Complex coefficients are permitted by the retained complex H interface. + +Consequently every gate type can be given **exactly six H factors**, separated by diagonal matrices: + +- Upper/lower shear: the six-H word above. +- Diagonal, including I: append three two-H identity words. +- Swap J: use J=(1/2)H diag(1,-1)H, then append two two-H identity words. +- FFT butterfly H diag(1,w), optionally scaled by 1/sqrt(2): append a two-H identity and a three-H identity. Here |w|=1. + +Combining neighboring diagonal matrices puts every word in one common alternating shape with six H factors and seven diagonals. Thus different gate types, zero parameters and inactive pairs can coexist in one simultaneous schedule without branches or padded data coordinates. + +## 3. Every matching can be routed to adjacent pairs + +For one local layer, assign its oriented pairs (i,j) to positions (2k,2k+1). Unpaired coordinates are assigned arbitrarily to the remaining positions, in pairs carrying diagonal or identity gates. This defines a permutation pi on exactly N coordinates. The layer equals + + P_pi^-1 (direct sum of adjacent-pair gates) P_pi. + +We use a Beneš network of 2m-1 switch layers. At each layer all switches join indices differing in one fixed bit; the bit sequence is + + 0,1,...,m-1,m-2,...,1,0. + +Here is a constructive proof of rearrangeability. Form a bipartite multigraph whose left vertices are input pairs {2j,2j+1} and whose right vertices are output pairs {2k,2k+1}. Input i supplies an edge to output-pair floor(pi(i)/2). Every vertex has degree two, so each component is an even cycle (parallel edges are allowed). Alternately color each cycle's edges 0 and 1. The first switch sends an input edge into the subnetwork of its color; the last switch sends that color to the required output parity. Each color class induces a permutation on N/2 pair indices. Recursively route both color subnetworks in parallel. This gives exactly the displayed bit sequence and uses no storage coordinates. + +The switch is either I or J. Both are obtained from the common two-H word + + S_epsilon=(1/2) H diag(1,epsilon) H, + epsilon=+1 for I, epsilon=-1 for J. + +Thus all axis-dependent switch controls are diagonal entries. A physical switch on bit k simply uses a simultaneous H call selecting bit k in each axis. No interstage wire shuffle is needed. If a supplier accepts only a contiguous selected-bit field, the existing paid bit router must gather that field and undo the gather; that is a supplier-interface cost, not an uncharged permutation. + +Each forward or inverse Beneš route uses 2(2m-1) H calls. The adjacent gates use six. Hence one original matching layer uses + + 2[2(2m-1)]+6 = 8m+2 + +H calls. The schedule is identical on every axis and every coarse-label block; only diagonal tables change. + +## 4. Tensor fusion is literal multiplication, not a sum expansion + +Let the compiled axis-a circuit be D_(a,L) H_(a,L) ... H_(a,1) D_(a,0), with the same number and arrangement of H factors. Kronecker multiplication gives + + tensor_a C_a + = (tensor_a D_(a,L)) (tensor_a H_(a,L)) ... + (tensor_a H_(a,1)) (tensor_a D_(a,0)). + +Every tensor of diagonals is a single diagonal scan. Every tensor of binary H factors is one simultaneous H call. This introduces neither a sum of 2^g branches nor a 2^g padding factor. The 2^g-dimensional Hadamard fibers are subsets of the N^g existing coordinates. + +An axis with no work at a particular original depth receives an identity layer, routed and compiled with the same fixed shape. Its intermediate H operations are still present, but the complete local word is I. Coarse-label variation is equally harmless: for each fixed label the identity above holds separately, and the whole data operator is a direct sum over labels. + +## 5. Uniform intermediate bounds and precision + +A completed switch route is a permutation and has norm one. Each partial switch word has absolute constant norm and inverse norm. Between original local circuit boundaries, at most one original gate word is partially executed. Its length is constant, and its scalar coefficients and inverses are poly(B); all different pairs act as a direct sum. Therefore + + max_prefix ||C_compiled,prefix||, + max_prefix ||C_compiled,prefix^-1|| <= poly(B) Lambda. + +The length O(D log N) does not require multiplying this polynomial bound afresh at every stage: completed original prefixes already have bound Lambda; completed route and padding words have their exact intended endpoints. The corresponding suffix bound is also polynomial when the original endpoint and inverse-prefix bounds are. + +For g tensor factors, the operator norms multiply, so their logarithms add. A microbox coarse-label direct sum takes the maximum over labels, not a product over labels. Hence polynomial local bounds give exp(O(g log N)); using only the rough original bound Lambda<=(2B)^D gives exp(O(gD log B)). Adding the original cell similarity contributes exp(sum Q_a). + +The number of compiled stages is polylog(p). Allocate each rounding or supplied-transform error below the final tolerance divided by this stage count and the prefix/suffix amplification. This costs O(g polylog(p)+sum Q_a) guard bits. Supplier-internal precision charges still must be those already guaranteed for the simultaneous H call; this compiler does not erase them. + +## 6. Product diagonals with constant amortized scan cost + +Fix one diagonal stage and one coarse label. Load the gN coefficients d_a(i) on a coefficient tape in axis order. Use base-N lexicographic enumeration with axis 0 fastest. Store suffix products + + P_j=product_(a=j)^(g-1) d_a(i_a), 1<=j=1) k/N^k = O(V), + +plus O(g) initialization. No division by d_a(i) is used, so this method also permits zero diagonal entries when an external application permits them. Every cached product is recomputed from current factors, never updated by a long chain of multiplicative ratios, so numerical drift does not depend on the number of records scanned. + +The same accounting works on a fixed number of tapes. Keep the coefficient-tape head normally in axis-0's N-record table. Rewinding that table costs O(N) once per N records. A depth-k carry visits the needed tables up to axis k and returns, moving O(kN) coefficient records. Summing over carries gives + + O(V sum_(k>=1) kN/N^k)=O(V) + +record moves. The suffix-product tape is accessed near its front except on carries, with the same geometric bound. Counter/address arithmetic costs only polylog(p) per record. This does not require g work tapes or a random-access lookup every record. + +The table loading cost is O(gN) records per coarse block and stage. For N>=2 it is O(N^g), including small g, and polynomial-time table construction is negligible per N^g block in the retained asymptotic regime g=p^beta tending to infinity. If tables are supplied rather than constructed locally, no additional assumption is needed. Both generation and loading costs must be charged once per visited coarse block, not silently once for the entire input. + +At O(p) working precision, O(1) amortized scalar multiplications per record and O(1) record movements cost O(V_bits polylog(p)) under the retained pointwise-arithmetic kernel, where V_bits counts bit volume. There is no g V_bits coefficient-generation term. + +## 7. Exact controls + +`check_tensor_compiler.py` constructs the network controls and all fixed words. `TENSOR-COMPILER-CONTROLS.json` records: + +- every permutation for N=2,4,8, including all 40,320 permutations at N=8; +- 100 random permutations each at N=16,32,64; +- exact symbolic checks of upper/lower zero and nonzero shears, diagonal, Swap and FFT words; +- an exact two-axis N=4 replay with different axis matchings and gate types: 16 data coordinates, all 256 input-output matrix entries, 18 simultaneous H calls, no dimension expansion. + +The finite controls support the constructive proofs; the cycle coloring, word identities, Kronecker identity and geometric scan sums establish the asymptotic statements. + +## 8. Cost substitution into the retained supplier + +Suppose a simultaneous H call on g selected bits, across V_bits bits of data and spectators, has paid cost O(V_bits g^(1-q) polylog(p)). Suppose also that any required bit-field gather/restore has cost O(V_bits g^(1-a_bit) polylog(p)). For D=polylog(p), N=poly(p), the compiler uses only polylog(p) such calls, and its diagonal scans cost O(V_bits polylog(p)). Thus the grouped local circuit costs + + O(V_bits [g^(1-q)+g^(1-a_bit)+1] polylog(p)). + +When a_bit>q and 00 for value in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + tests=[('endpoint_equality',dict(q=Q('0.001675462676808103522925'))), + ('zero_group_guard',dict(zeta=Q(0))), + ('zero_boundary_slack',dict(sigma_boundary=Q(0))), + ('old_per_axis_scan',dict(old_vd=True)), + ('unrecentered_full_cell_gauge',dict(full_cell_gauge=True))] + for name,kwargs in tests: + bad=calculate(**kwargs) + failed={k:v for k,v in bad['slacks'].items() if v<=0} + assert failed,name + controls[name]=failed + output=dict(status='PASS_CONDITIONAL_PHASE_FUSION_REVIEW', + scope='Local conditional transfer and exact arithmetic; inherited supplier and all-size interfaces remain assumptions. No upstream selected certificate is changed.', + review=dict(integrated_review='Root reviewed the complete standalone note, integrated proof chain, exact ledger, and independent cap/supplier/layout audits.', + independent_cap_gauge='CAPPED-GAUGE-AUDIT.md', + independent_cap_layout='CAPPED-SIGNOFF.md', + supplier_support='SUPPLIER-SUPPORT-AUDIT.md'), + **row,controls=controls) + (ROOT/'CAPPED-OUTER-ARITHMETIC.json').write_text(json.dumps(encode(output),indent=2)+'\n') + print(output['status']) + for name in ['epsilon','capped_block_power','beta_group','kappa']: + print(name,decimal(row['parameters'][name])) + print('minimum margin',decimal(min(row['margins'].values()))) + print('gain over selected',decimal(row['slacks']['improvement'])) + print('positive strict slacks',len(row['slacks'])) + print('controls',', '.join(controls)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/check_matrix_corollary.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/check_matrix_corollary.py new file mode 100644 index 0000000000..3f543803d7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/check_matrix_corollary.py @@ -0,0 +1,143 @@ +#!/usr/bin/env python3 +"""Exact corollary ledger, additionally assuming the supplied 9/4 theorem.""" +from fractions import Fraction as Q +from decimal import Decimal,localcontext +from pathlib import Path +import json + +ROOT=Path(__file__).resolve().parent + +def decimal(x): + with localcontext() as ctx: + ctx.prec=65 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def calculate(q=None,zeta=None,union_slack=None,old_vd=False, + full_cell_gauge=False,add_full_key_sort=False): + bit=Q('0.001733427039589844234749') + complex_=Q('0.001675462676808103522925') + q=complex_-Q(1,10**10) if q is None else q + a=(q+complex_)/2 + r=Q(1,10**6) + zeta=Q(1,10**8) if zeta is None else zeta + union_slack=Q(1,10**5) if union_slack is None else union_slack + tau_matrix=Q(19,8) + c=(3-tau_matrix)/2 + stop=Q(1,10**9) + cap=(Q(3,2)+q-(1+q)*(r+zeta)+union_slack)/(2+q) + eps=(1+q*(1-cap-r-zeta))/(1+q) + beta=1-cap-r-zeta + kappa=Q('0.001673360105405778') + selected=Q('0.001672660197076918') + first=Q('0.001672922349588871') + tau_bit,sigma=1-a,1-complex_ + internal=tau_bit+(1-stop)*max(sigma-tau_bit,Q(0)) + leaf=sigma+stop*(1-sigma) + lam=1-(q+a)/2 + key=beta+1-eps + margins=dict( + main_transform=eps*q, + global_coordinate_router=a, + fused_gaussian=1-eps+q*beta, + group_bit_router=1-eps+a*beta, + group_scans=1-eps+beta, + boundary_algebraic_matrix=c+cap-eps, + sparse_union_movement=1-eps-beta-Q(1,2)+cap, + polynomial_products=eps, + microbox_face_corrections=1-(eps+beta+3-8+2), + ) + if old_vd:margins['unreplaced_per_axis_scan']=1-eps + if add_full_key_sort: + margins['unnecessary_union_full_key_sort']=margins['sparse_union_movement']-key + slacks=dict( + actual_bit_above_effective=bit-a, + complex_above_effective=complex_-a, + complex_below_one_over_32=Q(1,32)-complex_, + q_positive=q, + complex_endpoint=complex_-q, + q_below_effective_bit=a-q, + stop_positive=stop, + stop_below_one=1-stop, + leaf_saving_above_effective_bit=(1-stop)*complex_-a, + q_below_internal=1-internal-q, + q_below_leaf=1-leaf-q, + lambda_above_tau=lam-tau_bit, + lambda_above_sigma=lam-sigma, + lambda_above_internal=lam-internal, + lambda_prime_above_lambda=1-q-lam, + lambda_prime_above_leaf=1-q-leaf, + epsilon_positive=eps, + epsilon_below_one=1-eps, + capped_block_above_band=cap-Q(1,2), + capped_block_below_cell=eps-cap, + group_positive=beta, + group_below_dimension=eps-beta, + support_router_span=1-beta, + group_gauge_power_slack=1-cap-r-beta, + actual_logarithmic_group_guard=1-cap-beta, + global_gamma_power_slack=1-eps-r, + actual_logarithmic_gamma_guard=1-eps, + alpha_logarithm_absorption=r, + group_key_fits_record=1-key, + membership_arithmetic_fits_record=1-beta-2*(1-eps), + sparse_union_fraction_decays=cap-beta-Q(1,2), + microbox_halo_volume=8-3-beta, + union_positive_slack=union_slack, + matrix_exponent_above_source_bound=tau_matrix-Q(9,4), + matrix_exponent_below_five_halves=Q(5,2)-tau_matrix, + short_record_fallback=eps-a, + improvement_over_selected=kappa-selected, + improvement_over_first_reviewed_transfer=kappa-first, + ) + if full_cell_gauge: + slacks['unrecentered_group_gauge']=1-eps-r-beta + slacks.update({name+'_above_kappa':value-kappa for name,value in margins.items()}) + assert margins['main_transform']==margins['fused_gaussian'] + assert margins['sparse_union_movement']==margins['main_transform']+union_slack + return dict(parameters=dict(a_bit=bit,a_eff=a,complex_saving=complex_,q=q, + r_spare=r,zeta=zeta,sigma_union=union_slack,stop=stop, + epsilon=eps,capped_block_power=cap,beta_group=beta, + matrix_exponent=tau_matrix,matrix_source_bound=Q(9,4), + c_boundary=c,kappa=kappa,selected=selected,first_reviewed_transfer=first, + lambda_=lam,lambda_prime=1-q,tau=tau_bit,sigma=sigma, + microbox_power=Q(8),halo_power=Q(3)),margins=margins,slacks=slacks) + +def encode(o): + if isinstance(o,Q):return dict(rational=str(o),decimal=decimal(o)) + if isinstance(o,dict):return {k:encode(v) for k,v in o.items()} + return o + +if __name__=='__main__': + row=calculate() + assert all(value>0 for value in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + tests=[('endpoint_equality',dict(q=Q('0.001675462676808103522925'))), + ('zero_group_guard',dict(zeta=Q(0))), + ('zero_union_slack',dict(union_slack=Q(0))), + ('old_per_axis_scan',dict(old_vd=True)), + ('unrecentered_full_cell_gauge',dict(full_cell_gauge=True)), + ('full_key_sort_on_union',dict(add_full_key_sort=True))] + for name,kwargs in tests: + bad=calculate(**kwargs) + failed={k:v for k,v in bad['slacks'].items() if v<=0} + assert failed,name + controls[name]=failed + output=dict(status='PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW', + scope='Additional matrix-conditioned corollary. Assumes supplied omega_C<=9/4 theorem and the effective algebraic stability transfer, as well as all first-transfer assumptions.', + review=dict(integrated='Root read the matrix corollary and independent layout signoff, reproduced the exact parameters and ledger, and replayed matrix source construction controls.', + matrix_source='PROOF-AND-COVERAGE.md', + effective_stability='ALGEBRAIC-STABILITY-INDEPENDENT.md', + independent_layout='MATRIX-LAYOUT-SIGNOFF.md'), + omitted_row=dict(name='optional full-coordinate-key union surcharge', + reason='The actual U_i complete-rectangle extraction, one-field transpose, and ordered merge do not sort union records. Sparse microhalo keys remain paid in their separate cost.'), + **row,controls=controls) + (ROOT/'MATRIX-COROLLARY-ARITHMETIC.json').write_text(json.dumps(encode(output),indent=2)+'\n') + print(output['status']) + for name in ['epsilon','capped_block_power','beta_group','kappa']: + print(name,decimal(row['parameters'][name])) + print('minimum margin',decimal(min(row['margins'].values()))) + print('gain over selected',decimal(row['slacks']['improvement_over_selected'])) + print('gain over first transfer',decimal(row['slacks']['improvement_over_first_reviewed_transfer'])) + print('positive strict slacks',len(row['slacks'])) + print('controls',', '.join(controls)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-CONTROLS.json new file mode 100644 index 0000000000..ed4b620d8a --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-CONTROLS.json @@ -0,0 +1,7 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "negative_and_zero_origin_instances": 45, + "finite_LDU_and_shift_entries": 750, + "scope": "Finite-origin algebra, exact origin-shift correction, upper inverse formula and simultaneous periodic shifts; bilateral limiting argument is analytic." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-LDU.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-LDU.md new file mode 100644 index 0000000000..5cad6ee19f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-LDU.md @@ -0,0 +1,131 @@ +# Bilateral Gaussian LDU and a fixed-width periodic inverse bound + +Independent audit of the parent's infinite-LDU observation. This is an analytic norm theorem. It does **not** assert that the bilateral factors already have a fully priced fast implementation. + +## Statement + +Let `1=1`, define on Z + +`G_ij=exp(-a[(rho*j-q_i)^2-beta_j^2])`. + +Let `r=exp(2a rho)` and `c=prod_(m>=1)(1-r^-m)>0`. Then G has an exact factorization on bounded bilateral sequences + +`G=L C U`, + +where L is unit lower triangular, U is unit upper triangular, C is diagonal, and all factors and their inverses are bounded. Moreover all six factors are invariant under simultaneous index translation by s. Their periodizations therefore factor the periodic Gaussian matrix N, and + +`||N^-1||_infinity <= c^-4 exp(a/2)/(1-exp(-a rho theta))^2`. + +In particular, at the fixed choice `a=1`, + +`||N^-1||_infinity < 16 theta^-2`. + +Any fixed a>=1, including a=4pi if an integer alpha>=2 is retained, gives a constant times theta^-2. No lower bound proportional to log(1/theta) on a is needed for this norm theorem. + +## 1. Finite-origin factorization and the shift correction + +Write `x_i=r^q_i`, `d_i=exp(-a q_i^2)`, and `e_i=exp(-a rho^2 i^2+a beta_i^2)`. Then `G_ij=d_i x_i^j e_j`, including negative indices, and `d_i x_i^i e_i=1`. + +For a finite square principal interval `[l,r0]`, the Vandermonde powers must begin at zero. The correct shifted row diagonal is **d_i x_i^l**, and the power is `j-l`. This removes a possible erroneous origin-dependent diagonal from the upper factor. + +Put + +`C_i(l)=prod_(l<=k=j`, + +`U^l_ij=(e_j/e_i) h_(j-i)(x_l,...,x_i)`, `j>=i`. + +Here h is the complete symmetric polynomial, h_0=1. Upper and lower inverses are + +`(U^l)^-1_ij=(-1)^(j-i)(e_j/e_i) e_(j-i)(x_l,...,x_(j-1))`, + +`(L^l)^-1_ij=(-1)^(i-j) exp(E_ij) C_i(l)/(C_j(l) D_ij)`, + +where + +`D_ij=prod_(j=j`, + +`|(U^l)^-1_ij|<=c^-1 exp(a/4) exp(-a rho theta(j-i))`, `j>=i`. + +Both use only `-1/2<=beta_k<1/2`. Thus their row norms are at most the corresponding constants divided by `1-exp(-a rho theta)`, uniformly in l and the finite size. + +The row sum of G is uniformly finite, since + +`G_i,i+h <= exp(a/4) exp(-a(rho*|h|-1/2)^2)`. + +The same positive Gaussian bound controls all finite-origin L and U factors. + +## 3. Letting the left origin tend to minus infinity + +For fixed i,j, C_i(l) converges to + +`C_i=prod_(k(i+s,j+s)`. Every displayed factor has the same invariance: ratios `x_k/x_i` are preserved, and the Gaussian/prefix expressions shift together. C_i is s-periodic. + +For an invariant matrix A with bounded absolute row sums, set + +`Pi(A)_ij=sum_(n in Z) A_(i,j+ns)`, `0<=i,j=e^2>7, and the elementary product bound gives `c>=1-sum_(m>=1)r^-m>5/6`. Also `1-exp(-rho theta)>=theta/2` for 04W, choose + +`B=min(Bcap, 2^floor(log2(R-W)))`, + +retain the next B vertices as an interior, then the next W vertices as boundary, and decrement R by B+W. Declare the final R<=4W vertices boundary. + +There are `O(L/Bcap+log Bcap)` interiors: while the cap is active, each step removes at least Bcap vertices; after it ceases to be active, the remaining length halves at each step. Every interior length is dyadic and at most Bcap. Every interior produced by the loop is longer than W; two successive interiors are separated by at least W retained coordinates. + +Total boundary count per original cell is + +`O(W(L/Bcap+log Bcap))`. + +Since original cell lengths are Theta(d), total boundary density is + +`O(W/Bcap + W log Bcap/d)=O((W/Bcap) polylog p)` + +when Bcap<=d. The same selected-input and target-output physical period cuts must be preserved. A clipped endpoint cell of O(W) length is wholly boundary. + +Interior elimination only fills between the adjacent boundary groups of one interior. Each group has O(W) vertices: inserted groups have W; an end remainder has at most 5W after joining the reserved end. Hence scalar Schur half-bandwidth remains O(W), independently of the cap and the number of pieces. No retained matrix coefficient is changed by this new partition. + +## 2. Narrow Gaussian: exact recentering of the existing rounded weights + +In one original phase cell, beta_i=theta*i-c and + +`log a_i=-a(1/4-beta_i^2)/theta`. + +For any two coordinates i,j in that same cell, + +`log(a_i/a_j)=a(i-j)(beta_i+beta_j)`. + +Since both beta values lie in [-1/2,1/2), + +`|log(a_i/a_j)|<=a|i-j|`. + +Thus a capped interior of at most Bcap vertices has ideal weight range at most exp(a Bcap), even when the absolute full-cell gauge has range exp(O(ad)). + +The actual matrix uses already-rounded positive weights ahat_i. Suppose, as ensured by the retained high-precision setup, + +`(1-epsilon) a_i <= ahat_i <= a_i`, with epsilon<=1/2. + +Then for an anchor m in the capped interior, + +`(1-epsilon) exp(-a Bcap) <= ahat_i/ahat_m` + +` <= exp(a Bcap)/(1-epsilon)`. + +Replacing ahat_i by `ahat_i/ahat_m` in both diagonal factors of + +`A=diag(ahat) Ghat diag(ahat)^-1` + +is **exact**. It changes neither A nor the fixed globally rounded H. These ratios must be formed in coefficient setup; the data circuit must not apply the full-cell diagonal and hope to cancel it later. One may anchor at the maximum ahat in the piece to make the direct diagonal contractive and its inverse at most `2 exp(a Bcap)`. + +The old setup provides absolute error below `4*2^-P` and ideal minimum at least exp(-a/(4theta)). It suffices that + +`4*2^-P exp(a/(4theta)) <= epsilon`. + +In the retained family a=Theta(log p), theta^-1=O(d), d=p^epsilon0 with fixed epsilon0<1, so ad=o(p). The existing P=Cp setup precision easily gives epsilon<=2^-cp. The **ad cost occurs in offline coefficient accuracy**, not in the runtime norm of the recentered diagonal. It is not multiplied by the number g of simultaneously executed axes. More explicitly, if the raw relative error is delta0<=4*exp(a/(4theta))*2^-P, then an anchored ratio has relative error at most 2delta0/(1-delta0). A g-fold product of anchored diagonals has error bounded by a polynomial factor times g*delta0*exp(O(g*a*Bcap)). Thus the needed precision is P>=p+O(ad+g*a*Bcap+log g+polylog p), rather than p+O(g*a*d). Both added power terms are o(p) in the proposed parameter family. + +## 3. Broad forward Gaussian: local gauges + +Use the target-grid notation from FORWARD-SMALL-A-AUDIT.md. With lambda=1/rho, b0=pi/alpha^2, original phase c, and an anchor m in the capped interior, + +`delta=(1-lambda)m-c`, `x=i-m`, `y=k-m`. + +The exact split is + +`Dcol(i)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(k)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`, + +with a fixed Toeplitz kernel `exp[-b0 lambda(i-k)^2]`. + +Throughout the original selected-target phase cell, `beta'_i=theta*i-rho*c` lies in [-1/2,1/2). Therefore + +`d(log Dcol)/di = -2 b0 lambda beta'_i`, + +`d(log Drow)/dk = 2 b0 lambda^2 beta'_k`. + +Their absolute slopes are at most b0 and b0 respectively. Both diagonal logarithmic ranges on the capped block are at most `b0 Bcap`. At the anchor, Dcol=1 and Drow=alpha^-1 exp(-b0 delta^2), so their absolute values also have logarithmic size `O(b0 Bcap+log alpha)`. + +If a globally rounded broad block was already defined using a different anchor, use exact constant ratios to recenter its two diagonals: divide the column diagonal by its value at m and multiply the row diagonal by the same value. Ideal changes of anchor differ only by reciprocal constants, since the derivatives above are independent of m. Uniform relative coefficient error changes the range by only O(1), just as in the narrow case. This preserves the already-defined rounded block exactly. Alternatively the broad block can be initially defined with the capped anchor, using one globally specified rounding construction. + +Holes remain boundary. The completed narrow and broad interior inverse bounds and the completed global/Schur norm bounds proved earlier do not depend on the cap: every new interior is still a contiguous principal block of the same original phase matrix. + +## 4. Group guard and circuit count + +Each capped narrow diagonal contributes at most `a Bcap+O(1)` to the logarithmic norm; the broad contribution `O(Bcap/alpha^2+log alpha)` is smaller. Shallow Toeplitz circuit amplification contributes polylog(p) per axis. The number of capped blocks per line does not multiply these bounds: they are disjoint direct-sum blocks and their circuits run in parallel, padding completed smaller circuits with identity gates. + +Consequently g simultaneously executed axes require + +`O(g a Bcap + g polylog p)` + +extra logarithmic precision. With `Bcap<=p^b`, `g<=p^beta`, a=Theta(log p), and beta+b=1-r-zeta with r,zeta>0, + +`g a Bcap=O(p^(1-r-zeta) log p)=o(p)`. + +In particular no remaining full-cell term g*a*d may be charged after exact recentering. Offline accuracy still needs to dominate ad=o(p), as above. + +## 5. Schur solves and whole-group ordering + +For the narrow matrix, strict row dominance is inherited by any interior elimination, so all completed Schur and inverse norms remain polynomial under the cap. + +For the broad matrix, completed Schur/inverse norms follow from the square identity-hole completion and polynomial principal-interior bounds in FORWARD-SMALL-A-AUDIT.md. To obtain uniform norms for a banded online factorization, eliminate **whole boundary groups** of W to O(W) vertices, rather than appealing without proof to scalar prefixes inside one group. + +After all interiors and a prefix of whole boundary groups are eliminated, the eliminated set is a direct sum of one physical contiguous prefix and disjoint future interiors: each remaining full group contains at least W uneliminated coordinates, blocking every width-W connection across it. The same holds after adding the next whole group. A physical prefix reduces, under selected/hole ordering, to the small-b normalized Gaussian block plus identity hole columns, so it has polynomial inverse norm. Each future interior also has polynomial inverse norm. Thus every required eliminated-set inverse is polynomial, uniformly in the number of pieces. + +At the final group on a full periodic line, use the already-established full periodic H inverse bound instead of a nonperiodic-prefix bound. For a clipped principal window the complement-gap condition excludes wrap edges, so the prefix statement applies directly. Each block pivot inverse is a principal block of the inverse after adding that group. Block Schur entries are bounded by `H+H H_EE^-1 H`, hence polynomial. There are only O(1) neighboring boundary groups in a band row. The resulting block-L and block-U norms are polynomial; their full inverses are polynomial as well, from `L^-1=U S^-1` and `U^-1=S^-1 L`. A scalar expansion inside a group is unnecessary: precompute its dense O(W)-square inverse and apply the block updates to batches by Strassen. + +This whole-group order prevents the subtle gap in which a partially consumed separator of fewer than W vertices permits a connection to the next interior. It preserves the claimed batched boundary exponent and is unaffected by capped subdivision. + +## Scope + +The capped local-gauge change is valid. The listed exact controls check carving, analytic slope identities and preservation of the already-rounded narrow matrix. The final recurrence arithmetic and tensor routing remain separate reviewed components. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/CAPPED-GAUGE-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/CAPPED-GAUGE-CONTROLS.json new file mode 100644 index 0000000000..6ea5c59eab --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/CAPPED-GAUGE-CONTROLS.json @@ -0,0 +1,11 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "integers and fractions.Fraction", + "checks": { + "partitions": 27216, + "narrow_slope_pairs": 6132, + "rounded_ratio_and_exact_reanchoring_pairs": 6132, + "broad_slope_pairs": 14754 + }, + "scope": "Capped dyadic carving, narrow exact log-slope, positive perturbed global weights, exact local reanchoring, and broad gauge log-slope. Final tensor schedule/ledger excluded." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/CYCLIC-SCAN-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/CYCLIC-SCAN-CONTROLS.json new file mode 100644 index 0000000000..eddd860c96 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/CYCLIC-SCAN-CONTROLS.json @@ -0,0 +1,5 @@ +{ + "status": "all exact assertions passed", + "cyclic_signed_scan_systems": 203, + "scope": "Two-pass cyclic affine recurrence against dense exact inversion; bilateral coefficient estimates are analytic." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/DYADIC-INTERIOR-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/DYADIC-INTERIOR-CONTROLS.json new file mode 100644 index 0000000000..e42eac6b93 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/DYADIC-INTERIOR-CONTROLS.json @@ -0,0 +1,6 @@ +{ + "status": "all exact assertions passed", + "cell_partitions": 80000, + "reachable_boundary_records": 6413386, + "scope": "Dyadic lengths, exact coverage, w-wide separation, logarithmic block count, and O(w) Schur boundary span." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-CELL-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-CELL-CONTROLS.json new file mode 100644 index 0000000000..c74f144886 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-CELL-CONTROLS.json @@ -0,0 +1,7 @@ +{ + "status": "all exact assertions passed", + "phase_cell_cases": 350, + "local_two_factorization_entries": 5184, + "selected_completion_entries": 16496, + "scope": "Exact target-grid coordinate conversion, local Toeplitz/Gaussian-normalized factorizations and selected block of square hole completion. Norm estimates are analytic." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md new file mode 100644 index 0000000000..0f7f80b3f7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md @@ -0,0 +1,109 @@ +# Broad forward Gaussian: finite-block and completed-matrix conditioning + +This note supports the phase-assembly branch. It distinguishes the broad forward map S from the narrow T used in N=C T D. Its conclusions concern conditioning and local circuit guards; the final simultaneous tensor layout remains that branch's obligation. + +Let the narrow parameter be `a=pi alpha^2=Theta(log p)`, `rho=t/s=1+theta`, and `lambda=1/rho`. The forward map is + +`S_(k,j)=alpha^-1 exp[-b0(j-lambda*k)^2]`, `b0=pi/alpha^2`. + +Equivalently it uses the small resampling parameter + +`b=b0 lambda^2=pi/(alpha^2 rho^2)`. + +## 1. The bilateral bound works at every positive parameter + +The finite and bilateral LDU arguments in BILATERAL-LDU.md use a>=1 only for their final numerical simplification of the product c. For any b>0, define + +`c_b=prod_(m>=1)(1-exp(-2b rho m))`. + +The same proof gives, for the periodic normalized matrix and for every finite contiguous principal block with q-gaps at least 1 and |beta_i|<=1/2, + +`||N_b^-1|| <= c_b^-4 exp(b/2)/(1-exp(-b rho theta))^2`. + +There is an explicit useful product bound. Expand the logarithm and sum nonnegative terms: + +`log(c_b^-1)=sum_(k>=1) 1/[k(exp(2b rho k)-1)]` + +` <=(1/(2b rho)) sum_(k>=1)1/k^2=pi^2/(12b rho)`. + +At `b=pi/(alpha^2 rho^2)`, this implies + +`c_b^-4 <= exp(pi alpha^2 rho/3)=exp(a rho/3)`. + +Since b*rho*theta=pi*theta/(alpha^2 rho)<1 for alpha>=2 and theta<=1, its reciprocal gap is at most `2alpha^2 rho/(pi theta)`. Thus every such normalized block inverse has norm polynomial in p when alpha^2=O(log p) and theta^-1 is polynomial in p. + +## 2. Actual target-grid phase-cell block + +Embed source j at target coordinate i=q_j. On an original phase cell q_j=j+c, every selected target coordinate i in that cell corresponds to j=i-c. A square interior of the forward target-grid matrix is therefore + +`K_(k,i)=alpha^-1 exp[-b0(i-lambda*k-c)^2]`. + +Here both i and k run over one contiguous selected-target interval inside that cell. Put + +`q'_k=k+rho*c`, `beta'_k=rho*k-q'_k=theta*k-rho*c`. + +At a selected coordinate k=q_j, beta'_k is exactly the original beta_j, hence lies in [-1/2,1/2). Also q' has consecutive gap 1. Integrality of q' was never needed in the finite LDU proof; monotone separation and the beta bound suffice. Consequently + +`K=alpha^-1 N_b diag(exp(-b*(beta'_i)^2))`, + +and + +`||K^-1|| <= alpha exp(3b/4) c_b^-4/(1-exp(-b rho theta))^2` + +` =O(alpha^5 rho^2 theta^-2 exp(pi alpha^2 rho/3))=poly(p)`. + +This controls the **completed** interior inverse. It avoids multiplying the large individual diagonal-gauge norms to estimate K inverse. Target holes should be retained as boundary vertices, so every interior coordinate has the asserted original beta bound. + +## 3. Square completion of the global forward map + +Add one identity column e_h for every target hole h, and feed zero into these extra columns. This makes an invertible square t-by-t matrix H_aug without changing the desired forward result. Permute selected q_j rows/columns first and holes second. In that order, + +`H_aug = [ alpha^-1 N_b D_b^-1 0 ; R I ]`, + +where `D_b=diag(exp(b beta_j^2))` and R consists of broad Gaussian rows at holes. A Gaussian lattice-sum bound gives `||R||=O(1)` after the alpha^-1 factor. Thus + +`H_aug^-1=[ alpha D_b N_b^-1 0 ; -R alpha D_b N_b^-1 I ]` + +has polynomial row norm. The matrix itself also has O(1) row norm. Gaussian truncation and sufficiently accurate structure-preserving coefficient rounding perturb both it and all required interior blocks by exponentially small amounts, preserving these polynomial bounds. + +If A is the block-diagonal interior matrix and S is the completed boundary Schur complement, then S inverse is the boundary principal block of H_aug inverse; hence its norm is polynomial. Also + +`||S||<=||H_BB||+||H_BI|| ||A^-1|| ||H_IB||=poly(p)`. + +This is enough for the completed Schur map. To justify a particular sequential block-LU implementation, its prefix factor norms still need to be checked. The natural physical-prefix elimination sets consist of one contiguous selected-source prefix plus disjoint future interiors, with identity hole columns. That geometry offers the same uniform contiguous-block inverse bound; it should be made explicit in the assembly proof. + +## 4. Exact local Toeplitz split and guard range + +Choose an anchor m inside the target cell, and write x=i-m, y=k-m, + +`delta=(1-lambda)m-c`, with |delta|<=1/2. + +Then the exact factorization is + +`K_(k,i)=Drow(y) G(y,x) Dcol(x)`, + +`G(y,x)=exp[-b0 lambda(x-y)^2]`, + +`Dcol(x)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(y)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`. + +Expansion gives exactly `-b0(delta+x-lambda*y)^2`; no approximate recentering is involved. On a phase cell of O(d) coordinates, the two diagonal logarithmic ranges are O(d/alpha^2). These are smaller than the O(alpha^2 d) narrow inverse gauges already paid by the proposed group size. + +## 5. SPD Toeplitz conditioning for the shallow circuit + +Let `gamma=b0 lambda=pi/(alpha^2 rho)`. The full Gaussian Toeplitz symbol has, by the Gaussian Poisson identity, + +`f(omega)=sqrt(pi/gamma) sum_(k in Z) exp[-(omega+2pi*k)^2/(4gamma)]`. + +For -pi<=omega<=pi, + +`sqrt(pi/gamma) exp[-pi^2/(4gamma)] <= f(omega) <= 1+sqrt(pi/gamma)`. + +Every finite principal Gaussian Toeplitz matrix has eigenvalues between these same bounds, by integrating the squared modulus of its coefficient polynomial against f. At alpha^2=O(log p), its condition number is polynomial in p. Truncation width O(sqrt(p/gamma)) and Cp-bit rounding perturb the matrix far less than its lower eigenvalue, so the actual symmetric rounded Toeplitz Ghat remains SPD with polynomial condition number. + +The low-displacement block-LU circuit of SHALLOW-TOEPLITZ.md extends from near identity to this SPD setting: every principal Schur complement has norm at most lambda_max and inverse norm at most lambda_min^-1, and ranks grow by the same additive displacement recurrence. Shear coefficients are polynomial in p; multiplying these bounds along O(log d) recursive levels gives exp(O(log^2 p)) partial-prefix growth. This calls for O(log^2 p) internal circuit guard bits per axis, rather than claiming the near-identity theorem's O(log p) unchanged. It is still below the group budget dominated by the local Gaussian diagonals. + +Dyadic interior subdivision from LOG-A-PHASE-SCHUR-AUDIT.md applies here too, using the broader width O(sqrt(p alpha^2)). It keeps target holes and separators as boundary, adds only polylogarithmic boundary volume, and lets every interior use a dyadic shallow circuit without 2^g padding. + +No numerical final exponent follows from these conditioning statements alone. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md new file mode 100644 index 0000000000..3bcf490a8c --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md @@ -0,0 +1,140 @@ +# Audit: logarithmic Gaussian width retains the phase-cell Schur solver + +This audit checks the analytic hypotheses of the retained rad20 phase-cell solver at `a=pi alpha^2=Theta(log p)`, `theta=Theta(1/d)`, even though `a theta` tends to zero. It also audits the shallow Toeplitz circuit algebra and identifies the actual matrix that must be precomputed. It does not certify the final tensor layout or multiplication exponent. + +Read inputs: `references/semantic-bulk/rad20/reports/review-bulk-resampling.md`, `review-phase-cell-inverse.md`, `downstream-phase-cell-inverse.md`, and `work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md`. + +## 1. The sharp elementary diagonal-dominance estimate + +Let `rho=1+theta`, `beta_i=rho*i-q_i`, and let the Gaussian row coefficient at integer offset h be + +`n_i,h=exp(-a[(rho*h+beta_i)^2-beta_(i+h)^2])`. + +The h=0 coefficient is 1. Set `g_i=q_i-q_(i-1)` and `g_(i+1)=q_(i+1)-q_i`, each in {1,2}. For the nearest coefficients, + +`n_i,-1=exp(-a*g_i*(2rho-2beta_i-g_i))`, + +`n_i,+1=exp(-a*g_(i+1)*(2rho+2beta_i-g_(i+1)))`. + +If g_i=1, the first exponent is exactly `1+2theta-2beta_i`; if g_i=2, that gap forces `beta_i=1/2-theta`, so the upper exponent is at least `1+2theta+2beta_i`. Consequently + +`n_i,-1+n_i,+1` + +` <=exp(-a(1+2theta-2beta_i))+exp(-a(1+2theta+2beta_i))` + +` <=exp(-2a theta)+exp(-2a(1+theta))`. + +The last step is convexity in beta on [-1/2,1/2]. The two nearest directions cannot both be charged as exp(-2a theta). + +For |h|>=2, + +`(rho*h+beta_i)^2-beta_(i+h)^2 >= h^2-|h|`, + +so their entire two-sided tail is below `4 exp(-2a)` for a>=1. Therefore + +`||N-I||_infinity <= exp(-2a theta)+5 exp(-2a)`. + +This includes nonzero periodic diagonal aliases in the tail; no alias is silently omitted from this bound. + +For p>=101, theta>=1/(4p), and `a>=4 log(16p)`, + +`1-exp(-2a theta) >= 1-exp(-1/(2p)) >=1/(4p)`, + +`5exp(-2a)<=5/(16p)^8<1/(8p)`. + +Hence N has row diagonal-dominance margin at least `1/(8p)` in the following robust sense: after deleting its tiny nonzero diagonal aliases, its diagonal is 1 and its off-diagonal row sum is at most `1-1/(8p)`. In particular the fixed-width inverse theorem is not needed to supply the retained strict row dominance. + +Choose the legal integer alpha by + +`alpha=ceil(sqrt((4/pi)log(16p)))`. + +Then a has the required lower bound and is O(log p). The old sufficient condition `u>=ceil(log2(8p))` need not be reused: the direct estimate above replaces that prerequisite. + +## 2. Actual rounded H, interior matrices, and guard range + +Use exactly the retained structure: + +- `w=ceil(sqrt(16p/u))+2`, u=alpha^2=a/pi; +- cell weights `a_j=exp(-a(1/4-beta_j^2)/theta)`; +- one globally fixed downward dyadic Toeplitz kernel `Ghat_delta` for |delta|<=w, zero outside, with Ghat_0=1; +- positive downward dyadic weights ahat_j; +- within a physical cell, define H_jk **exactly** as `ahat_j Ghat_(j-k)/ahat_k`; +- across cells, use the retained downward Gaussian entries; set H_jj=1. + +Thus the actual interior block being inverted is + +`A_cell=diag(ahat) Ghat_cell diag(ahat)^-1`. + +It is not the unrounded Gaussian matrix, and its retained truncated Ghat is not silently replaced by the infinite Gaussian kernel. + +The weight range is + +`0<=-log a_j<=a/(4theta)=O(a d)` + +under theta between 1/(4d) and 1/(2d-1). The old eventual condition u*d<=p holds whenever d=p^epsilon, epsilon<1, and u=O(log p). It bounds the old rational perturbation calculation by the same `2^(O(p)) eta` allowance. The stronger range `O(a d)` should be used for grouped tensor precision, rather than charging O(p) independently for every axis. + +Choose the coefficient precision high enough that `||H-N||<1/(16p)`, including the Gaussian truncation, omitted diagonal aliases and weight-ratio rounding. The retained bound `p*2^(16p+7)*eta+2^(-46p)` at eta=2^(-32768p) is more than sufficient. Then H has diagonal 1, row gap at least `1/(16p)`, row norm below 2, and inverse norm at most 16p. All clipped principal windows inherit the same row gap. The physical period cut must remain a cell boundary, as in the retained bulk review. + +## 3. Schur complement bounds without an a*theta>1 assumption + +A scalar elimination step on a strictly row-diagonally-dominant matrix cannot decrease its minimum absolute row gap. For pivot k, + +`gap_i(new) >= gap_i(old)+(|a_ik|/|a_kk|) gap_k(old)`. + +The full absolute row sum also cannot increase, since the added update row has off-pivot sum less than |a_kk|. Induction applies regardless of the signs of filled entries. Since the diagonal begins real positive and the correction magnitude is below it, the surviving diagonal remains positive. + +Consequently the exact Schur complement + +`S=H_BB-H_BI H_II^-1 H_IB` + +has row gap at least `1/(16p)`, row norm below 2, positive diagonal, and + +`||S^-1||_infinity<=16p`. + +Principal interior inverses also have norm at most 16p. Rounding S by a row perturbation below 1/(32p) preserves gap at least 1/(32p) and inverse norm at most 32p. A row has O(w) nonzeros; the retained P-grid readily meets this target. These are polynomial bounds on arbitrary signed inputs, with no factor equal to the number of cells. + +The boundary geometry is unchanged: keep first/last w vertices of each physical cell, yielding O(s*w/d) boundary vertices and circular half-bandwidth at most 2w. Schur fill only joins the two end groups of one cell, while original cross-cell entries join neighboring end groups. Actual inverse and LU coefficient tables must be precomputed for this H/S, not imported from the old parameter values. + +Thus the retained banded/cyclic boundary solver's gap-dependent proof remains applicable with fixed constants enlarged. The proposed batched boundary arithmetic still needs its own data-layout and Strassen error audit, but there is no analytic conditioning obstruction at logarithmic a. + +## 4. Apply the shallow circuit directly to Ghat inverse + +Because a is eventually larger than 4pi, the symmetric Toeplitz matrix Ghat_cell satisfies + +`max(||Ghat_cell-I||_1,||Ghat_cell-I||_infinity)<1/8`. + +Downward rounding only decreases this off-diagonal sum. The shallow full-Toeplitz theorem therefore applies directly to Ghat_cell. Reverse the resulting invertible circuit and invert its gates to obtain Ghat_cell inverse. This avoids the two-branch Gohberg-Semencul difference entirely. The interior inverse is: + +1. multiply by diag(ahat)^-1; +2. apply the reversed shallow circuit for Ghat_cell; +3. multiply by diag(ahat). + +Its only superpolynomial individual scale comes from the **local** weight range O(a d). Completed A_cell inverse has polynomial norm by the strict row gap. Across g axes, the uncompleted local weight contribution is at most exp(O(g*a*d)); choosing `g<=p/(d polylog p)` with sufficient logarithmic slack keeps the total data precision O(p). The shallow circuit contributes polynomial-in-d amplification per axis, hence only O(g log d) additional logarithmic range. This is a guard calculation, not a complete simultaneous-layer routing proof. + +## 5. Independent shallow-Toeplitz algebra audit and a length caveat + +The parity-split triangular identity, block Schur displacement formulas, Stein-generator telescoping identity, and bounded-prefix induction in SHALLOW-TOEPLITZ.md check algebraically. In particular the parity children run in parallel, avoiding the earlier erroneous sequential-depth recurrence. The full block-LU recursion also runs its two children in parallel, with O(log N) displacement rank after O(log N) levels. + +That document currently states its circuit theorem for dyadic N. Actual cell interiors are not generally dyadic. Padding every tensor axis to its next power of two would be invalid. A plausible exact extension has been communicated to its author: split an arbitrary N into balanced sizes differing by at most one; a near-square rectangular Toeplitz dirty shear is a square Toeplitz shear plus one extra row/column rank-one shear. The latter can be implemented by a binary tree of unitary 2-coordinate reductions and its inverse, without clean ancilla. The rectangular Stein identity telescopes as before. This extension needs its own written proof/controls before it is used as an established interface. + +No new final kappa is asserted by this audit. + +## 6. Dyadic interior subdivision resolves the length caveat + +No arbitrary-length shallow FFT theorem is actually needed. Refine the interior/boundary partition inside each original physical phase cell of length L as follows. Keep its first and last w vertices as boundary. Starting with the remaining interval of length R=L-2w: + +- while R>4w, choose the largest power of two B<=R-w; +- mark the next B vertices as one interior, the following w vertices as boundary, and replace R by R-B-w; +- once R<=4w, mark all remaining vertices as boundary. + +Every interior now has exactly dyadic length. If R>4w, then `B>(R-w)/2`, so the next remainder is less than R/2. There are `O(1+log(L/w))` interiors and the total boundary count in the cell is `O(w(1+log(L/w)))`. The final boundary group, including the initially reserved last w vertices, has size at most 5w; all inserted separators have size w. + +The width-w matrix H cannot connect two different new interiors: their separating boundary gap has w vertices, so their nearest physical coordinates differ by w+1. Each new interior is still a principal block of the SAME globally fixed Toeplitz-similar cell matrix. Its weights retain the original cell gauge; there is no new rescaling or rerounding at a subdivision boundary. + +Eliminating one dyadic interior fills only between its immediately neighboring boundary groups. Each such group has O(w) vertices. Original H edges also connect only the same group or neighboring groups; an interior has length greater than w in the loop above. Thus, in inherited physical boundary order, the new Schur complement has half-bandwidth O(w), not O(w log d). Only its number of rows grows by O(log d): + +`n_boundary=O((s/d) w log d)`. + +Clipped endpoint cells of size O(w) may be declared wholly boundary, adding only O(w) rows; preserve the original physical period cut. This is the same endpoint convention as the retained bulk review. + +The boundary solve and coupling work therefore gain only polylogarithmic factors. All dyadic interior blocks use the established SHALLOW-TOEPLITZ theorem directly, without padding any tensor axis and without requiring a non-dyadic Fourier circuit. This resolves the length caveat in section 5 for the intended phase-cell application; it does not prove the separate simultaneous tensor-layer routing claim. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-SCHUR-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-SCHUR-CONTROLS.json new file mode 100644 index 0000000000..301815a2c5 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-SCHUR-CONTROLS.json @@ -0,0 +1,11 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "checks": { + "nearest_exponent_checks": 147618, + "far_exponent_checks": 442854, + "schur_eliminations": 468, + "schur_entries": 19032 + }, + "scope": "Nearest/far Gaussian exponent inequalities and signed/asymmetric Schur row-gap preservation; the transcendental row-sum estimate and tensor transfer are analytic." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-CONTROLS.json new file mode 100644 index 0000000000..df9fc504c1 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-CONTROLS.json @@ -0,0 +1,13 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "checks": { + "instances": 63, + "inverse_entries": 2424, + "tile_entries": 4110, + "suffix_outputs": 1377, + "exponent_bounds": 2424, + "upper_entries": 2424 + }, + "scope": "Nonperiodic Newton L inverse and U inverse algebra; exact signed cross-tile data application; no runtime or tensor-fusion certificate." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-FACTOR.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-FACTOR.md new file mode 100644 index 0000000000..35af6c7b99 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-FACTOR.md @@ -0,0 +1,180 @@ +# A fast, stable application of the general Newton lower factor + +Status: a proved one-dimensional **nonperiodic lower-factor** algorithm and exact controls. It does not by itself implement the entire periodic inverse. The upper factor has a polynomial norm bound but its fast normalization/fusion remains to be supplied. No final multiplication exponent is claimed. + +## 1. Exact Gaussian LDU + +Let `rho=1+theta`, `q_i=floor(rho*i+1/2)`, `beta_i=rho*i-q_i`, `x_i=r^q_i`, `r=exp(2a rho)>1`, and `d_i=exp(-a q_i^2)`, for `0<=i=j`, zero otherwise. + +The ordinary monic Newton expansion of `X^j` gives exactly + +`G=L diag(C_i) U`, + +where `L=diag(d_i) T diag(d_i)^-1`, `L_ii=U_ii=1`, and + +`(U^-1)_ij=(-1)^(j-i) (e_j/e_i) e_(j-i)(x_0,...,x_(j-1))`, + +`e_j=exp(-a rho^2 j^2+a beta_j^2)`. + +The pivot calculation is exact: + +`d_i P_i(x_i) e_i = C_i`. + +Define + +`A_i=d_i x_i^i C_i/prod_(k<=i)x_k` + +` =C_i exp(-a beta_i^2-a rho^2 i+2a rho sum_(k<=i) beta_k)`. + +Divided differences give the exact inverse lower entries + +`(L^-1)_ij=(-1)^(i-j) (A_i/A_j)/D_(i,j)`, + +`D_(i,j)=prod_(j=1)(1-r^-m)`. Because `q_(i+1)-q_i>=1`, + +`c<=C_i,D_(i,j)<=1`. + +For `i>=j`, direct summation gives + +`log |A_i/A_j| <= -log c+a/4-a rho theta(i-j)`. + +Consequently + +`||L^-1||_infinity <= c^-2 exp(a/4)/(1-exp(-a rho theta))`. + +For the upper factor, compare every elementary-symmetric term with the product of the largest `j-i` variables. The sorted deficits form a partition, hence + +`1<= e_(j-i)(x_0,...,x_(j-1))/prod_(i<=k=-1/2` gives + +`||U^-1||_infinity <= c^-1 exp(a/4)/(1-exp(-a rho theta))`. + +Also `||diag(C_i)^-1||<=c^-1`. For `a>=1`, `r>=e^2`, so `c` has an absolute positive lower bound. For `a=O(log(1/theta))`, both factor norms are polynomial in `theta^-1`. These bounds apply to **arbitrary signed/complex input vectors**, not only correlated outputs of another factor. + +## 3. One stable scan plus a local correction + +Let + +`D_j^+=prod_(j=0`, + +`0<=1-H_i(x_j)<=sum_(k>i)x_j/x_k <= r^(-h-1)/(1-r^-1)`. + +The sum of discarded correction entries for `h>=W` is therefore at most + +`c^-2 exp(a/4) r^(-W-1)/[(1-r^-1)(1-exp(-a rho theta)/r)]`. + +Thus `W=O((p+a+log(1/theta))/a+1)` makes the omitted correction less than `2^-p` in operator norm. At `a=O(log(1/theta))`, `theta>=p^-O(1)`, this is `O(p/a+1)`. + +Partition the line into pages of W indices. Compute the exact correction on each diagonal page and each immediately lower adjacent page, and omit it elsewhere. Every omitted entry has `h>=W`; every retained tile has O(W) rows/columns. A diagonal page is recursively split into ordered cross tiles plus its diagonal. Only O(log W) tree levels are required. The adjacent-page rectangle can use the tile algorithm below directly. Subtracting the scan's contribution on those same tiles gives exactly the desired retained correction. + +## 4. The bounded-polynomial cross-tile algorithm + +Let every column in J precede every row in `I=[l,b]`, and suppose both sets lie in an index interval of length O(W). Put + +`z_j=x_j/x_l` (j in J), `v_k=x_l/x_k` (l=0) M_m Z^m=sum_j w_j/(1-z_j Z)=Q(Z)/P(Z)`. + +Build a balanced tree with leaf pairs `(P,Q)=(1-z_j Z,w_j)` and merge by + +`P=P_left P_right`, + +`Q=Q_left P_right+Q_right P_left`. + +Truncate to the needed degree at each node. Since `sum z_j<=1/(r-1)`, all product coefficient l1 norms are at most `exp(1/(r-1))<2`; numerator norms are at most twice the absolute input weight sum. The power-series reciprocal of P has l1 norm at most `c^-1`. Standard fast power-series inversion, or doubling the geometric series of `1-P` (whose l1 norm is <1/2 for r>=e^2), is therefore stable with polynomially bounded intermediates. Product-tree complexity is soft-O(|J|+|I|) scalar operations. + +### Fast suffix-polynomial evaluation + +The second operation is the transpose of multiplication by the suffix products. Split `[l+1,b]` at m. Right-hand output rows use the moment vector restricted to their degree range. For left rows, replace the moment vector by + +`M'_h=sum_k [Z^k]prod_(m=1; at a=1 this is `<16 theta^-2`. The independent exact negative-origin controls are BILATERAL-CONTROLS.json. +2. **Complete one-dimensional data application:** LOWER-FACTOR.md supplies the lower factor with logarithmic guard overhead. UPPER-DATA-ALGORITHM.md supplies the upper factor and a direct periodic implementation with `O(s p polylog p)` data cost, shared per-axis coefficient preprocessing, and **Cp internal precision**. This is stated in the target `s>>poly(p)` regime. Exact controls are LOWER-CONTROLS.json, UPPER-DATA-CONTROLS.json, and CYCLIC-SCAN-CONTROLS.json. +3. **Length-preserving upper merge:** UPPER-LIFTING.md proves the Newton merge and bounded exact shear algebra. UPPER-MERGE-CONTROLS.json checks its degree indexing and factorization. +4. **Still open:** a realization of the normalized Hankel/Toeplitz steps with only logarithmic internal guard bits, and a bounded-depth width-preserving tensor circuit. Neither the analytic norm theorem nor the one-dimensional time bound establishes those, and no improved final multiplication exponent is claimed. + +Replay all finite controls with Python 3: + +``` +python3 work/gaussian_fast_ultra/check_lower.py +python3 work/gaussian_fast_ultra/check_upper_merge.py +python3 work/gaussian_fast_ultra/check_bilateral.py +python3 work/gaussian_fast_ultra/check_upper_data.py +python3 work/gaussian_fast_ultra/check_cyclic_scan.py +``` + +The package uses no third-party source result beyond the elementary Newton polynomial identities stated and proved by their constructions. It retains the parent Gaussian notation and does not modify the frozen upstream manuscript. + +Focused phase-fusion audit added later: + +- `LOG-A-PHASE-SCHUR-AUDIT.md`: verifies strict row dominance at a=Theta(log p), polynomial Schur bounds, actual rounded Toeplitz interiors, and dyadic interior subdivision with only a logarithmic increase in boundary count. `LOG-A-SCHUR-CONTROLS.json` and `DYADIC-INTERIOR-CONTROLS.json` contain independent exact checks. +- `FORWARD-SMALL-A-AUDIT.md`: handles the distinct broad forward Gaussian S via b=pi/(alpha^2 rho^2), gives polynomial finite-block/completed-matrix inverse bounds, the exact target-grid chirp factorization, and the SPD Toeplitz circuit guard estimate. `FORWARD-CELL-CONTROLS.json` checks the coordinate and matrix identities. + +These are interface components for the parent's focused partial-fusion proposal. They do not independently certify its full tensor schedule or final exponent. + +- `CAPPED-GAUGE-AUDIT.md` independently validates capped dyadic interiors and exact reanchoring of the already-rounded narrow matrix, with local log range O(aB) rather than O(ad). It also validates the broad O(B/alpha^2) gauge and whole-boundary-group solve order. `CAPPED-GAUGE-CONTROLS.json` contains 27,216 partition checks and exact narrow/broad gauge controls. The capped final exponent ledger remains a separate assembly artifact. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-ALGORITHM.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-ALGORITHM.md new file mode 100644 index 0000000000..f07f2c68c7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-ALGORITHM.md @@ -0,0 +1,166 @@ +# A complete upper-inverse data algorithm with linear precision overhead + +Status: the construction below closes the one-dimensional **data application** of the nonperiodic upper inverse in `O(s p polylog p)` bit operations, after fixed per-axis coefficient preprocessing, using `C p` internal precision for an absolute constant C. It does not obtain logarithmic internal guard bits and does not prove tensor fusion. The lower-factor algorithm in LOWER-FACTOR.md does have logarithmic guard overhead. + +Let `p` be the required output precision, `a>=1`, `theta>=p^-O(1)`, and `a=O(p)`. In the target family a is fixed or logarithmic and every line length s grows faster than any fixed power of p. Preprocessing depending on the axis is shared by all its lines, as allowed by the parent reduction; it is not charged anew for every line. The coefficient words actually read by one data application have total `O(s p polylog p)` bits. + +## 1. Upper inverse is a stable suffix scan plus bandwidth O(p/a) + +Use the column diagonal `e_i=exp(-a rho^2 i^2+a beta_i^2)`. Define + +`B_j=e_j prod_(0<=k=i`, write + +`C_i,j=e_(j-i)(x_0,...,x_(j-1))/prod_(i<=k0. Put `d=m-i`, + +`gamma_i=(U^-1)_i,m`, + +`tau_j=(-1)^(j-m)(B_j/B_m)`. + +An exact Laurent-polynomial coefficient identity gives + +`(U^-1)_ij=gamma_i tau_j sum_(h>=0) (alpha_(d+h)/alpha_d) beta_j,h`. + +The signs are already in gamma and tau; the remaining coefficient sum is nonnegative. Both gamma and tau are polynomially bounded, with the same exponential-distance primary bounds as before. + +To verify the identity, write X=x_m Z and factor + +`P_j(X)=X^m prod_(k=1)(1+r^-k) <= 2/c`. + +A truncated product/sum tree computes M_0,...,M_R in `O((|J|+R) polylog p)` scalar operations. At a split, its exact rule is `S=S_left+P_left S_right`, with P a fixed product. Intermediate norms are bounded by a constant times the absolute weighted input sum, independently of tile length. No large Gaussian diagonal occurs in this step. + +## 3. Exponentially small high degrees + +The partition bound on alpha and beta gives + +`alpha_(d+h)/alpha_d <= c^-1 r^(-d*h-h(h+1)/2)`, + +`beta_j,h <= c^-1 r^(-h(h-1)/2)`. + +Thus the h-term, after summing input weights, is bounded by + +`c^-2 r^(-d*h-h^2) sum_j |tau_j u_j|`. + +Group row distances dyadically: D<=d<2D. For that group retain only + +`R=min(|J|, O(sqrt(P)), O(P/D))`, `P=(p+a+log(1/theta))/a`, + +with sufficiently large absolute constants. If |J| is smaller, the polynomial is already exact. Otherwise all omitted terms sum to less than `2^-p` after allowing the polynomial gamma/tau factors. There are O(log p) row groups. + +## 4. Fast normalized Hankel application, with Cp internal bits + +The remaining map for a row group is + +`y_d=sum_(0<=h<=R) (alpha_(d+h)/alpha_d) M_h`. + +Split the d-range into consecutive blocks of length at most R (or the whole group if shorter). For a block beginning at d0, divide all alpha coefficients used by alpha_d0. Then the numerator is one reversed ordinary convolution of that coefficient segment with M. Divide output d by alpha_d/alpha_d0. + +The decisive height bound is local, not global. If k>=0, comparing dominant products and using q-gaps in {1,2} gives + +`c r^(-2*d0*k-k(k+1)) <= alpha_(d0+k)/alpha_d0 <= c^-1`. + +Here the lower bound is needed only when the coefficient exists; every queried alpha index is <=m. If alpha vanishes because its degree exceeds m, store exact zero and never divide by it. Since d0=O(D), k=O(R), and `D R+R^2=O(P)`, every nonzero coefficient ratio and every divisor in this block lies between `exp(-O(aP))` and an absolute constant. Thus **Cp internal precision** suffices for these convolutions and the final divisions. + +The number of scalar operations for one row group is `O(D polylog p)`; its separate bounded-prefix computation costs `O((|J|+R) polylog p)`. Summing the O(log p) row groups gives `O((|I|+|J|) polylog p)` scalar operations per ordered tile. Recursing through one W-page or processing adjacent pages therefore costs `O(W p polylog p)` bits. Over all pages the data cost is + +`O(s p polylog p)`. + +The Gaussian factors gamma and tau can be evaluated from bounded dominant-product ratios rather than raw e_i or B_i. Precomputed alpha tables can similarly be stored with dominant-product gauges. Every alpha segment actually read by the data algorithm has length O(|I|+|J|) per row-group and tile level; this accounts for the coefficient tape scans in the same bound. + +A conservative preprocessing implementation may compute every required alpha table from O(poly(p)) nearby nodes separately, and can therefore cost `s poly(p)` per axis. This is a one-time axis cost, not an unpriced per-line cost. In the parent regime it is negligible compared with the full tensor data volume. A uniformly quasi-linear preprocessing theorem is not asserted here. + +## 5. Complete nonperiodic and periodic application + +Combining this upper map with the lower map and the bounded diagonal gives the complete nonperiodic inverse `G^-1=U^-1 C^-1 L^-1` in `O(s p polylog p)` data time, at Cp internal precision, with polynomial operator norms. Requested final error is achieved by a fixed additional factor in precision plus O(log p+log(1/theta)) endpoint guard bits. + +For the periodic matrix one may either implement the bilateral factors in BILATERAL-LDU.md or use a finite boundary correction. The latter needs only two nonperiodic inverse data applications: + +- Truncate nonzero Gaussian aliases to the two corner blocks of width `b=O(sqrt((p+log(1/theta))/a)+1)`, obtaining `K=G+UV` and a row-norm tail smaller than `2^-Cp`. +- U inserts O(b) boundary coordinates and V reads a bounded O(b)-by-O(b) corner matrix. +- Precompute `S^-1`, `S=I+V G^-1 U`. Its norm is polynomial because `S^-1=I-V K^-1 U` and the periodic inverse norm plus the tiny discarded tail bounds K inverse. +- Apply `x=G^-1 z`, `w=S^-1 Vx`, and return `x-G^-1 Uw`. + +The boundary data work is polynomial in p. For the target line lengths s growing faster than every fixed polynomial in p, it is negligible relative to sp. The updated matrix in the Woodbury inverse identity is K, not the base G. The periodic Gaussian tail and final correction error are bounded using the fixed-a theorem in BILATERAL-LDU.md. + +## What is still not established + +The normalized Hankel step uses Cp working precision because its local alpha divisors may be exp(-Cp). It has not been converted to a circuit whose every intermediate operator has polynomial norm. Therefore this result does not meet the stronger **logarithmic guard at every internal stage** requirement needed by the proposed full-dimensional fusion. It supplies a complete one-dimensional data algorithm and a precise remaining normalization target; it supplies no improved final kappa. + +## 6. Direct periodic implementation from the bilateral factors + +The bilateral theorem also permits the Woodbury step above to be omitted. This is cleaner in the target regime `s>>W`. + +For the bilateral lower inverse, use + +`C_i=prod_(ki)(1-x_i/x_k)`. + +The same lower formula and scan/correction split hold, with the defining products now bilateral. The adjacent A ratio is still given by the beta formula in LOWER-FACTOR.md. For the upper inverse, use elementary polynomials in the infinite prefix `x_k, k=1-exp(-a)`. Intermediate partial products and scan norms have the polynomial bounds already proved. This supplies a stable two-pass scan on the original s records. + +Retain the correction only for one-sided distances below W on the integer cover of the circle. With `s>>W`, each retained coefficient belongs to its diagonal page or the single adjacent cyclic page; virtual negative or beyond-s indices are generated by periodic q/beta formulas. The same local tile algorithms apply. Periodization cannot increase the row norm of any discarded tail, so all the previous error estimates remain valid. + +In the upper tile algorithm, alpha_d is now the coefficient of the infinite small-root prefix + +`prod_(k=d+O(q/a+1)`: a contribution involving an omitted node has partition deficit at least K-d, and summing the corresponding partition tail gives `O(r^(-(K-d)/2))` relative error (bound the tail by r to half the negative deficit times the convergent partition generating product). For the O(W)-degree tables at q=Cp, this is O(W+p/a) nodes, independent of s. Normalized table storage and data access are unchanged. + +Thus, after fixed per-axis coefficient preprocessing, the bilateral factorization itself gives the **periodic** inverse data application in `O(s p polylog p)` time with Cp working precision, in the target regime `s>>poly(p)`. It needs neither a large raw Vandermonde nor a boundary Schur system. The unresolved logarithmic-internal-guard and tensor-fusion qualifications remain exactly as stated above. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-CONTROLS.json new file mode 100644 index 0000000000..0d9410b6a8 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-CONTROLS.json @@ -0,0 +1,12 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "checks": { + "instances": 75, + "suffix_recurrences": 2715, + "tile_entries": 6413, + "signed_prefix_tree_and_hankel_outputs": 1833, + "alpha_ratio_bounds": 10008 + }, + "scope": "Algebra, signs, degree indexing, suffix-tail recurrence, and signed blocked upper data application. Cp-bit stability and asymptotic data cost are analytic, not certified by finite tests." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-LIFTING.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-LIFTING.md new file mode 100644 index 0000000000..0985ad16cd --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-LIFTING.md @@ -0,0 +1,57 @@ +# The upper factor has a length-preserving Newton merge + +This is a structural advance, not yet the missing fast normalized FFT implementation. It concerns the nonperiodic factor from LOWER-FACTOR.md. + +Write `d_i=exp(-a rho^2 i^2+a beta_i^2)` in this note (this is the column diagonal called `e_i` in LOWER-FACTOR.md). The upper inverse is + +`(U^-1)_ij=(d_j/d_i) [X^i] prod_(k=m and h=j-i, the logarithm of the last Gaussian factor is at most + +`a/4-a rho^2 h^2-a rho(2rho-1)h-2a rho^2(i-m)h`. + +Thus D inverse also has row norm `O(exp(a/4))`, and the shear B D inverse has polynomial norm. These statements hold on arbitrary signed/complex records. + +## Remaining realization problem + +The algebra is length preserving and the exact blocks have bounded norm. Directly executing their diagonal-Toeplitz-diagonal presentation is not yet safe for full-dimensional tensor fusion: the raw d_i ratio can have logarithmic range `Theta(a(r-l)^2)`. A local FFT using that presentation therefore reintroduces an O(p) guard for blocks of length sqrt(p/a). + +A plain unit triangular Toeplitz map can be implemented by recursive Toeplitz shears and in-place cyclic/negacyclic Fourier transforms, as suggested by the parent investigation. To close this upper-factor route, that primitive needs a version for the **Gaussian-conjugated** Toeplitz blocks above whose intermediate norms stay polynomial, or a different bounded lifting factorization of them. The final exact block norms alone do not prove the intermediate FFT normalization claim. + +The numerator Sylvester merge in the lower-factor moment tree remains a separate lifting problem; the one-product Newton merge here avoids that two-product merge for the upper factor only. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-MERGE-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-MERGE-CONTROLS.json new file mode 100644 index 0000000000..7e851839f8 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-MERGE-CONTROLS.json @@ -0,0 +1,6 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "normalized_node_merges_and_shear_factorizations": 286, + "scope": "Algebra and degree indexing, not bounded FFT realization or tensor fusion." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_bilateral.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_bilateral.py new file mode 100644 index 0000000000..b2f527e951 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_bilateral.py @@ -0,0 +1,45 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from pathlib import Path +import json +from check_lower import matmul, inv, elem + +def pp2(n):return F(2)**n +def prod(v): + z=F(1) + for x in v:z*=x + return z +def complete(xs,n): + c=[F(1)]+[F(0)]*n + for x in xs: + for k in range(1,n+1):c[k]+=x*c[k-1] + return c[n] +def case(s,t,l,n): + q=lambda i:(2*t*i+s)//(2*s) + x=lambda i:pp2(2*s*t*q(i)) + e=lambda i:pp2(-t*t*i*i+(t*i-s*q(i))**2) + G=lambda i,j:pp2(-(t*j-s*q(i))**2+(t*j-s*q(j))**2) + C=lambda i,o:prod(1-x(k)/x(i) for k in range(o,i)) + L=lambda i,j,o:G(i,j)*prod(1-x(k)/x(i) for k in range(o,j))/C(j,o) if i>=j else F(0) + U=lambda i,j,o:e(j)/e(i)*complete([x(k) for k in range(o,i+1)],j-i) if j>=i else F(0) + ids=list(range(l,l+n)); LL=[[L(i,j,l) for j in ids] for i in ids];UU=[[U(i,j,l) for j in ids] for i in ids] + CC=[[C(i,l) if i==j else F(0) for j in ids] for i in ids] + assert matmul(matmul(LL,CC),UU)==[[G(i,j) for j in ids] for i in ids] + UI=inv(UU) + for ii,i in enumerate(ids): + for jj,j in enumerate(ids): + assert L(i+s,j+s,l+s)==L(i,j,l) + assert U(i+s,j+s,l+s)==U(i,j,l) + assert G(i+s,j+s)==G(i,j) + if j>=i: + assert UI[ii][jj]==(-1)**(j-i)*e(j)/e(i)*elem([x(k) for k in range(l,j)],j-i) + return n*n + +def main(): + instances=entries=0 + for s,t in [(2,3),(3,4),(3,5),(4,5),(4,7)]: + for l in [-5,-2,0]: + for n in [3,4,5]:entries+=case(s,t,l,n);instances+=1 + out={'status':'all exact assertions passed','arithmetic':'fractions.Fraction','negative_and_zero_origin_instances':instances,'finite_LDU_and_shift_entries':entries,'scope':'Finite-origin algebra, exact origin-shift correction, upper inverse formula and simultaneous periodic shifts; bilateral limiting argument is analytic.'} + Path(__file__).with_name('BILATERAL-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) +if __name__=='__main__':main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py new file mode 100644 index 0000000000..d6f5a14cb2 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py @@ -0,0 +1,73 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from pathlib import Path +import json + +def p2(n):return F(2)**n + +def carve(L,W,cap): + if L<=6*W:return [('B',0,L)] + ps=[('B',0,W)];pos=W;R=L-2*W + while R>4*W: + B=min(cap,1<<((R-W).bit_length()-1)) + ps.append(('I',pos,pos+B));pos+=B + ps.append(('B',pos,pos+W));pos+=W;R-=B+W + if R:ps.append(('B',pos,pos+R));pos+=R + ps.append(('B',pos,L)) + return ps + +def main(): + c={'partitions':0,'narrow_slope_pairs':0,'rounded_ratio_and_exact_reanchoring_pairs':0,'broad_slope_pairs':0} + for W in [1,2,3,5,9,17]: + for power in range(2,11): + cap=1<=W for x,y in zip(ins,ins[1:])) + budget=(L+cap-1)//cap+cap.bit_length()+10 + assert len(ins)<=budget + assert sum(b-a for typ,a,b in ps if typ=='B')<=W*budget + c['partitions']+=1 + for s in range(3,31): + for K in range(1,min(s,6)): + rho=F(s+K,s);theta=rho-1;lam=1/rho + q=[(2*(s+K)*i+s)//(2*s) for i in range(s)] + cells={} + for i in range(s):cells.setdefault(q[i]-i,[]).append(i) + # a/log(2)=4*s*K makes every original narrow weight dyadic. + exps=[4*(K*i-s*(q[i]-i))**2-s*s for i in range(s)] + aa=[p2(e) for e in exps];eta=p2(-2*s*s-100) + ah=[x-(1+i%4)*eta for i,x in enumerate(aa)] + for phase,inds in cells.items(): + for start in range(0,len(inds),3): + block=inds[start:start+3];m=block[0] + for i in block: + for j in block: + assert abs(exps[i]-exps[j])<=4*s*K*abs(i-j) + c['narrow_slope_pairs']+=1 + assert F(1,2)*p2(-4*s*K*abs(i-j))<=ah[i]/ah[j]<=2*p2(4*s*K*abs(i-j)) + kernel=F(2,3) if i!=j else F(1) + before=ah[i]*kernel/ah[j] + after=(ah[i]/ah[m])*kernel/(ah[j]/ah[m]) + assert before==after + c['rounded_ratio_and_exact_reanchoring_pairs']+=1 + # Broad gauges on target coordinates in the original physical phase cell. + target=[q[i] for i in inds];anchor=target[len(target)//2] + delta=(1-lam)*anchor-phase + def col(k): + x=k-anchor + return -((1-lam)*x*x+2*delta*x) + def row(k): + y=k-anchor + return lam*(1-lam)*y*y+2*delta*lam*y-delta*delta + for i in target: + for j in target: + assert abs(col(i)-col(j))<=abs(i-j) + assert abs(row(i)-row(j))<=abs(i-j) + c['broad_slope_pairs']+=1 + out={'status':'all exact assertions passed','arithmetic':'integers and fractions.Fraction','checks':c,'scope':'Capped dyadic carving, narrow exact log-slope, positive perturbed global weights, exact local reanchoring, and broad gauge log-slope. Final tensor schedule/ledger excluded.'} + Path(__file__).with_name('CAPPED-GAUGE-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) +if __name__=='__main__':main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_cyclic_scan.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_cyclic_scan.py new file mode 100644 index 0000000000..7f8ebba6af --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_cyclic_scan.py @@ -0,0 +1,25 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from pathlib import Path +import json +from check_lower import matmul,inv,eye + +def scan(lam,u): + P=F(1);Q=F(0) + for a,b in zip(lam,u):P=a*P;Q=a*Q+b + prev=Q/(1-P);out=[] + for a,b in zip(lam,u):prev=a*prev+b;out.append(prev) + return out + +count=0 +for n in range(2,31): + for offset in range(7): + lam=[F((-1)**(i+offset)*(i%3+1),5+i%4) for i in range(n)] + u=[F((-1)**i*(2*i+offset+1),i+1) for i in range(n)] + A=eye(n) + for i,a in enumerate(lam):A[i][(i-1)%n]-=a + expected=[r[0] for r in matmul(inv(A),[[x] for x in u])] + assert scan(lam,u)==expected + count+=1 +out={'status':'all exact assertions passed','cyclic_signed_scan_systems':count,'scope':'Two-pass cyclic affine recurrence against dense exact inversion; bilateral coefficient estimates are analytic.'} +Path(__file__).with_name('CYCLIC-SCAN-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_dyadic_interiors.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_dyadic_interiors.py new file mode 100644 index 0000000000..9a970cb94e --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_dyadic_interiors.py @@ -0,0 +1,38 @@ +#!/usr/bin/env python3 +from pathlib import Path +import json + +def parts(L,w): + if L<=2*w:return [('B',0,L)] + out=[('B',0,w)];pos=w;R=L-2*w + while R>4*w: + B=1<<((R-w).bit_length()-1) + out.append(('I',pos,pos+B));pos+=B + out.append(('B',pos,pos+w));pos+=w;R-=B+w + if R:out.append(('B',pos,pos+R));pos+=R + out.append(('B',pos,L)) + return out +cases=edges=0 +for w in range(1,41): + for L in range(1,2001): + ps=parts(L,w) + assert ps[0][1]==0 and ps[-1][2]==L + assert all(a[2]==b[1] for a,b in zip(ps,ps[1:])) + ins=[x for x in ps if x[0]=='I'] + assert all((b-a)&(b-a-1)==0 for _,a,b in ins) + assert all(y[1]-x[2]>=w for x,y in zip(ins,ins[1:])) + assert len(ins)<=2+max(0,(max(1,L//w)).bit_length()) + assert sum(b-a for t,a,b in ps if t=='B')<=w*(8+max(1,L//w).bit_length()) + # For each new interior, all boundary neighbors reachable by an original band edge + # lie in its two adjacent retained blocks. Eliminating it can fill only their union. + labels=[-1]*L;bid=0 + for typ,a,b in ps: + if typ=='B': + for i in range(a,b):labels[i]=bid;bid+=1 + for _,a,b in ins: + ns=[labels[i] for i in list(range(max(0,a-w),a))+list(range(b,min(L,b+w))) if labels[i]>=0] + if ns:assert max(ns)-min(ns)<=10*w + edges+=len(ns) + cases+=1 +out={'status':'all exact assertions passed','cell_partitions':cases,'reachable_boundary_records':edges,'scope':'Dyadic lengths, exact coverage, w-wide separation, logarithmic block count, and O(w) Schur boundary span.'} +Path(__file__).with_name('DYADIC-INTERIOR-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_forward_cell.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_forward_cell.py new file mode 100644 index 0000000000..b91b7a6bad --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_forward_cell.py @@ -0,0 +1,40 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from pathlib import Path +import json + +def p2(k):return F(2)**k +cases=entries=completion=0 +for s in range(3,22): + for t in range(s+1,min(2*s,s+5)+1): + rho=F(t,s);lam=1/rho;q=[(2*t*j+s)//(2*s) for j in range(s)] + cells={} + for j,i in enumerate(q):cells.setdefault(i-j,[]).append(i) + for c,inds in cells.items(): + m=inds[len(inds)//2];delta=F((t-s)*m-t*c,t) + assert abs(delta)<=F(1,2) + for k in inds: + betak=(rho-1)*k-rho*c + assert -F(1,2)<=betak=1+2*th-2*b + assert upper>=1+2*th+2*b + counts['nearest_exponent_checks']+=2 + for h in [-4,-3,-2,2,3,4]: + assert (rho*h+b)**2-beta(i+h)**2>=h*h-abs(h) + counts['far_exponent_checks']+=1 + # Both signs and asymmetric rows: no positivity hypothesis on filled off-diagonals. + for n in range(3,15): + for seed in range(6): + A=[[F(1) if i==j else F(((-1)**(i+j+seed))*((7*i+3*j+seed)%11+1),40*n) for j in range(n)] for i in range(n)] + original_gap=min(gap(A,i) for i in range(n)); original_norm=norm(A) + while len(A)>2: + k=(seed+len(A))%len(A);idx=[i for i in range(len(A)) if i!=k] + B=[[A[i][j]-A[i][k]*A[k][j]/A[k][k] for j in idx] for i in idx] + assert min(gap(B,i) for i in range(len(B)))>=original_gap + assert norm(B)<=original_norm + assert all(B[i][i]>0 for i in range(len(B))) + counts['schur_eliminations']+=1;counts['schur_entries']+=len(B)**2 + A=B + return counts +if __name__=='__main__': + out={'status':'all exact assertions passed','arithmetic':'fractions.Fraction','checks':run(),'scope':'Nearest/far Gaussian exponent inequalities and signed/asymmetric Schur row-gap preservation; the transcendental row-sum estimate and tensor transfer are analytic.'} + Path(__file__).with_name('LOG-A-SCHUR-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_lower.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_lower.py new file mode 100644 index 0000000000..1260c26dd1 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_lower.py @@ -0,0 +1,92 @@ +#!/usr/bin/env python3 +"""Exact finite controls for LOWER-FACTOR.md; no floating point.""" +from fractions import Fraction as F +from math import prod +import json +from pathlib import Path + +def matmul(A,B): + return [[sum((a*b for a,b in zip(row,col)),F(0)) for col in zip(*B)] for row in A] +def eye(n): return [[F(i==j) for j in range(n)] for i in range(n)] +def inv(A): + n=len(A); B=[list(r)+list(e) for r,e in zip(A,eye(n))] + for k in range(n): + v=B[k][k]; assert v; B[k]=[x/v for x in B[k]] + for i in range(n): + if i!=k: + v=B[i][k]; B[i]=[x-v*y for x,y in zip(B[i],B[k])] + return [r[n:] for r in B] +def poly_mul(a,b): + out=[F(0)]*(len(a)+len(b)-1) + for i,x in enumerate(a): + for j,y in enumerate(b): out[i+j]+=x*y + return out +def newton_poly(xs): + out=[F(1)] + for x in xs: out=poly_mul(out,[-x,F(1)]) + return out +def elem(xs,k): return (-1)**k*newton_poly(xs)[len(xs)-k] +def suffix_pull(vs,M): + """All =i)(1-v_k Z),M>, exact D&C transpose tree.""" + if not vs: return [M[0]] + if len(vs)==1: return [M[0]-vs[0]*M[1],M[0]] + mid=len(vs)//2; left,right=vs[:mid],vs[mid:] + P=[F(1)] + for v in right: P=poly_mul(P,[F(1),-v]) + LM=[sum((P[k]*M[h+k] for k in range(len(P))),F(0)) for h in range(len(left)+1)] + return suffix_pull(left,LM)[:-1]+suffix_pull(right,M[:len(right)+1]) + +def run(): + count={'instances':0,'inverse_entries':0,'tile_entries':0,'suffix_outputs':0,'exponent_bounds':0,'upper_entries':0} + for s in range(3,12): + for t in range(s+1,2*s+1): + rho=F(t,s); q=[(2*t*i+s)//(2*s) for i in range(s)] + beta=[rho*i-q[i] for i in range(s)]; x=[F(2)**k for k in q] + # Arbitrary exact rational diagonal tests all identities independent of Gaussian values. + d=[F(3)**(-q[i]*q[i]) for i in range(s)] + C=[prod((1-x[k]/x[i] for k in range(i)),start=F(1)) for i in range(s)] + D=lambda i,j:prod((1-x[j]/x[k] for k in range(j+1,i+1)),start=F(1)) + A=[d[i]*x[i]**i*C[i]/prod(x[:i+1],start=F(1)) for i in range(s)] + L=[[F(0)]*s for _ in range(s)] + for i in range(s): + for j in range(i+1): + L[i][j]=d[i]/d[j]*prod((x[i]-x[k] for k in range(j)),start=F(1))/prod((x[j]-x[k] for k in range(j)),start=F(1)) + LI=inv(L) + for i in range(s): + for j in range(i+1): + assert LI[i][j]==(-1)**(i-j)*A[i]/A[j]/D(i,j) + count['inverse_entries']+=1 + h=i-j + exponent=-beta[i]**2+beta[j]**2-rho**2*h+2*rho*sum(beta[j+1:i+1],F(0)) + assert exponent<=F(1,4)-rho*(rho-1)*h + count['exponent_bounds']+=1 + # Newton U inverse coefficients and V=L D U, with row diagonal removed. + V=[[x[i]**j for j in range(s)] for i in range(s)] + T=[[L[i][j]*d[j]/d[i] for j in range(s)] for i in range(s)] + U=matmul(inv(T),V) + for i in range(s): + pivot=prod((x[i]-x[k] for k in range(i)),start=F(1)) + U[i]=[v/pivot for v in U[i]] + UI=inv(U) + for i in range(s): + for j in range(i,s): + assert UI[i][j]==(-1)**(j-i)*elem(x[:j],j-i) + count['upper_entries']+=1 + # Every adjacent-cut tile, arbitrary signed inputs, exact moment/suffix algorithm. + u=[F((-1)**j*(2*j+1),j+1) for j in range(s)] + for l in range(1,s): + b=min(s-1,l+3); J=list(range(max(0,l-4),l)); I=list(range(l,b+1)) + z=[x[j]/x[l] for j in J]; v=[x[l]/x[k] for k in range(l+1,b+1)] + w=[(-1)**j*A[l]/A[j]/D(b,j)*u[j] for j in J] + M=[sum((wj*zj**m for wj,zj in zip(w,z)),F(0)) for m in range(len(I))] + pulled=suffix_pull(v,M) + for i,got in zip(I,pulled): + got*=(-1)**i*A[i]/A[l] + expect=sum((LI[i][j]*u[j] for j in J),F(0)) + assert got==expect + count['suffix_outputs']+=1; count['tile_entries']+=len(J) + count['instances']+=1 + return count +if __name__=='__main__': + out={'status':'all exact assertions passed','arithmetic':'fractions.Fraction','checks':run(),'scope':'Nonperiodic Newton L inverse and U inverse algebra; exact signed cross-tile data application; no runtime or tensor-fusion certificate.'} + p=Path(__file__).with_name('LOWER-CONTROLS.json');p.write_text(json.dumps(out,indent=2)+'\n'); print(json.dumps(out,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_data.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_data.py new file mode 100644 index 0000000000..ad06229d84 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_data.py @@ -0,0 +1,79 @@ +#!/usr/bin/env python3 +"""Exact checks of the upper suffix recurrence, tile identity, and blocked data algorithm.""" +from fractions import Fraction as F +from pathlib import Path +import json +from check_lower import poly_mul, elem + +def prod(xs): + z=F(1) + for x in xs:z*=x + return z + +def add(a,b): + out=[F(0)]*max(len(a),len(b)) + for i,x in enumerate(a):out[i]+=x + for i,x in enumerate(b):out[i]+=x + return out + +def prefix_sum(roots,weights): + """sum weights[j] product_{k=0 + assert delta<=2*F(4)**(-(j-i+1)) + out['suffix_recurrences']+=1 + u=[F((-1)**j*(j+2),j+1) for j in range(s)] + for m in range(1,s): + l=max(0,m-4);r=min(s,m+5);I=list(range(l,m));J=list(range(m,r)) + aa=[elem([x/xs[m] for x in xs[:m]],h) for h in range(m+1)] + tau=[(-1)**(j-m)*BB[j]/BB[m] for j in J] + gamma={i:U(i,m) for i in I} + roots=[xs[m]/xs[k] for k in range(m,r-1)] + moments=prefix_sum(roots,[tau[k]*u[j] for k,j in enumerate(J)]) + R=len(moments)-1 + for i in I: + d=m-i + for j in J: + b=[elem([xs[m]/xs[k] for k in range(m,j)],h) if h<=j-m else F(0) for h in range(R+1)] + v=sum((aa[d+h]/aa[d]*b[h] for h in range(R+1) if d+h<=m),F(0)) + assert gamma[i]*tau[j-m]*v==U(i,j) + out['tile_entries']+=1 + # Blocks of length two for normalized Hankel convolution. + for d0 in range(1,m-l+1,2): + B=min(2,m-l-d0+1) + a=[aa[d0+k]/aa[d0] if d0+k<=m else F(0) for k in range(B+R)] + cv=poly_mul(a,list(reversed(moments))) + for off in range(B): + d=d0+off;i=m-d + got=gamma[i]*cv[off+R]/(aa[d]/aa[d0]) + expected=sum((U(i,j)*u[j] for j in J),F(0)) + assert got==expected + out['signed_prefix_tree_and_hankel_outputs']+=1 + for d in range(1,m+1): + for k in range(m-d+1): + rat=aa[d+k]/aa[d] + assert F(1,2)*F(4)**(-2*d*k-k*(k+1))<=rat<=2 + out['alpha_ratio_bounds']+=1 + out['instances']+=1 + return out +if __name__=='__main__': + out={'status':'all exact assertions passed','arithmetic':'fractions.Fraction','checks':run(),'scope':'Algebra, signs, degree indexing, suffix-tail recurrence, and signed blocked upper data application. Cp-bit stability and asymptotic data cost are analytic, not certified by finite tests.'} + Path(__file__).with_name('UPPER-DATA-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_merge.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_merge.py new file mode 100644 index 0000000000..3dda87e144 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_merge.py @@ -0,0 +1,43 @@ +#!/usr/bin/env python3 +from check_lower import F, poly_mul, newton_poly, matmul, eye, inv +from pathlib import Path +import json + +def local(xs,ds,l,r): + n=r-l;A=[[F(0)]*n for _ in range(n)] + for j in range(l,r): + P=newton_poly(xs[l:j]) + for k,p in enumerate(P):A[k][j-l]=ds[j]*p/ds[l+k] + return A + +def main(): + count=0 + for n in range(2,13): + xs=[F(2)**(i+i//3) for i in range(n)] + ds=[F(3)**(-i*i) for i in range(n)] + for l in range(n-1): + for r in range(l+2,n+1): + m=(l+r)//2;n1=m-l;n2=r-m + P=newton_poly(xs[l:m]);M=eye(r-l) + for i in range(l,r): + for j in range(m,r): + k=i-l-(j-m) + M[i-l][j-l]=ds[j]/ds[i]*P[k] if 0<=k4L` and `gL/N=o(1)`. +Write `b=log2 N`, `k=gb`, and `K0=O(g(h+b)+log g)=O(gh)` for a correction key. +Require `K0^2=O(B)` or retain the explicit extra metadata term below. + +The complete cost is + +``` +O(V [ (gb)^(1-a) polylog p + + (gb)^(1-q) polylog p + + polylog p + + (gL/N) (K0 + g polylog p) ]) ++ polynomial(p) setup, +``` + +plus `O((gL/N)V K0^2/B)` if metadata arithmetic is not absorbed by records. +All scans, duplication, sparse sorts, local transposes and final output +order are included. In particular, there is no gV layout term. + +## 1. One support-limited gather + +Within the active group, gather its g low b-bit fields into a contiguous +suffix using the support-at-most-2gb extension proved in +`SUPPORT-ROUTER.md`. The remaining active coarse bits remain in their +possibly scrambled order. Every coarse tile is now one contiguous block +of N^g complete R-payloads. No other active coarse bit is put into a new +stable order, and no spectator coordinate is gathered. + +The initial and inverse gather together cost +`O(V (gb)^(1-a) polylog p)`. The explicit permutation maps each physical +coarse tile name back to its logical vector; that small map is used only +for tile setup and sparse correction metadata. + +## 2. Interior outputs from ordinary cyclic convolution + +Flatten a tile by `phi(x)=sum_i x_i N^(g-1-i)` into Q=N^g slots. Put the +product kernel at offsets `psi(z)=sum_i z_i N^(g-1-i) mod Q` in an ordinary +one-dimensional cyclic convolution kernel. These offsets are distinct: +reduce a putative collision modulo N, use `2L= p^eta`, `g -> infinity`; alternatively a supplied sequential dense +kernel generator is an explicit acceptable input. + +If every tile coordinate satisfies `L<=x_ij: [0,N). +``` + +These 2g rectangles are disjoint and contain every discarded output once. +Their required unwrapped input rectangles have intervals + +``` +ij: [-L,N+L). +``` + +Coordinates outside `[0,N)` are taken from the correct neighboring coarse +tile, including periodic wrap in each actual axis. All input dimensions +are less than 2N; no assumption of independent full 2N padding is used. + +The total copied input volume is exactly + +``` +M = 6L V/N^g * sum_{j=0}^{g-1} N^j (N+2L)^(g-1-j) + = 3V [ (1+2L/N)^g - 1 ] + = O((gL/N)V). +``` + +The corrected output volume is +`V [1-(1-2L/N)^g] = O((gL/N)V)`. The formulas count corner duplication +exactly. An individual corner may have many destinations; its total traffic +is already included in M. No bound by worst-case 2^g multiplicity is charged +to all records. + +## 4. One full scan generates all correction copies + +For each original core tile, recover its logical coarse coordinates once. +Scan its local coordinate tuple with a binary odometer. Maintain flags for +membership in the four fixed edge intervals `[0,L)`, `[0,2L)`, +`[N-2L,N)` and `[N-L,N)`. Only coordinates changed by the odometer update +their flags. The number of coordinate updates over a whole tile is +`Q(1+1/N+...)<2Q`. Comparisons cost O(log N), and counters/head resets use +the retained amortized stream-counter convention. Maintain a count of +active near-boundary flags. A nonexceptional record takes no scan through +the g flags. + +For an exceptional record, enumerate candidate first-boundary axes from +the flag tape and all valid neighboring-tile choices. The exact choices +are the ones implemented in `sparse_copies` in `check_halo.py`: earlier +axes stay in the original tile; the selected face axis has its own or one +adjacent tile; each later axis has its own tile plus an adjacent tile only +when within L of that edge. Every emitted record carries + +``` +(j, side, destination coarse tile, unwrapped local coordinates). +``` + +These are K0=O(gh) bits. Precompute the plus/minus-one forms of each logical +coarse coordinate once per tile. A copy's logical neighbor is generated +by selecting those forms, then encoding the known scrambled coarse order. +All carries, periodic wraps and changed tile order are explicit. Using +ordinary binary arithmetic gives O(K0^2) safe metadata work per copy. + +The full scan costs O(V polylog p). The exceptional flag scans and address +work cost O(M K0^2/B+M) in bit-volume units when R includes the B-bit record. +Their setup per coarse tile is absorbed by its N^g payload. This is not an +unpaid nonlinear boundary-signature evaluation on every payload bit. + +## 5. Sparse radix gather and fixed-tape local processing + +Within each *earlier-coordinate prefix fiber*, stable-radix sort the copies +by the displayed key, with unwrapped local coordinates in ordinary tensor +order. Only the g active axes occur in the key; all following spectator +coordinates remain one R-block payload. A radix pass scans each complete +header and payload, splits into two bucket tapes, concatenates, and resets +the same fixed set of tapes. Since K0=O(B), each pass costs O(M); K0 passes +cost O(M K0). No simultaneous open tape per tile, face or corner is needed. + +Sorted copies form the exact input rectangles of section 3, consecutively. +Process one rectangle at a time. For each axis, expose its current local +coordinate field, apply the length-polynomial one-dimensional convolution, +and crop L positions from each end. Thus its length changes from its input +interval to the corresponding output interval. Other axes remain unchanged +until their own pass. This yields the tensor product of the restricted +one-dimensional maps, hence the exact desired face outputs. + +Every current axis has length at most N+2L=poly(p). Its mixed-radix transpose +is the retained short-field construction: pad **only that field** to its +next power of two temporarily, move its O(log p) bits using O(log p) linear +block interchanges, remove the temporary padding, and use the correct new +field length on return. The temporary factor below two is removed before +the next axis. A rectangle therefore costs its volume times +`O(g polylog p)` for movement and unconditional polynomial-length arithmetic. +Summing over rectangles costs `O(M g polylog p)`. Signed/complex dyadic +convolution can use sign splitting and ordinary Kronecker packing with +O(p+log N)=O(p) guard bits, followed by the unconditional quasilinear integer +multiplier; no improved supplier is used on this sparse path. + +After cropping, attach each output's physical coarse-tile and flat inner +address. Their keys are distinct because the face output partition is +disjoint. A second sparse radix sort orders them as holes in the retained +core-output stream. One merge scan replaces exactly those holes. The inverse +support-limited gather restores the original axis-major stream, including +the initially displaced coarse coordinates. All metadata is discarded. + +## 6. Parameters for the proposed partial fusion + +Use `d=p^epsilon`, `h=Theta(p/d)`, `g=p^(1-epsilon-zeta)` with fixed positive +zeta and epsilon>5/6. Suppose `L=O(p polylog p)` and take the dyadic side +`N=2^ceil((3/2)log2 p)`. The actual S_i eventually exceed every polynomial, +so N divides them and each periodic window is shorter than S_i. + +Then `K0=O(p^(2(1-epsilon)-zeta) polylog p)` and `K0^2=o(p)`. The two sparse +cost multipliers satisfy + +``` +(gL/N) K0 = O(p^(5/2-3epsilon-2zeta) polylog p) = o(1), +(gL/N) g = O(p^(3/2-2epsilon-2zeta) polylog p) = o(1). +``` + +All halo work is therefore O(V polylog p), below the desired +`O(V g^(1-a) polylog p)` routing term. The core convolution is +`O(V g^(1-q) polylog p)` since log N=Theta(log p). + +The bounded-gauge condition cited in `PARTIAL-FUSION.md` remains payable: +with `a0=Theta(log d)` and theta=Theta(1/d), its variation is + +``` +g [a0 theta N + a0/theta] + = O(p^(5/2-2epsilon-zeta) polylog p + + p^(1-zeta) polylog p) + = o(p) +``` + +for epsilon>5/6 and fixed zeta>0. Thus the larger polynomial tile does not +consume the O(p)-bit guard budget. This checks a layout/precision inequality; +it does not establish the missing Gaussian convolution factorization. + +## Diagonal and factorization boundary + +The theorem also applies to a fixed or polylogarithmic composition of these +translation-invariant convolutions and diagonal factors if every diagonal +scan/factor page has a proved sequential O(V polylog p) implementation and +all normalized tile-core and tile-halo values use O(p) bits. A tile-dependent +common scalar gauge may multiply every input to that tile and divide its +outputs; boundary copies must use the **destination** tile's gauge. No +enormous diagonal is formed and later canceled numerically by assertion. + +A concrete sufficient diagonal implementation is available for +`D(x)=product_i d_i(x_i)`. Supply each one-axis factor page in coordinate +order, with all normalized prefix products bounded to O(p) bits. Traverse +the rectangle by nested coordinate loops and retain the g successive prefix +products on work tape. Multiply by the innermost factor once per record; +outer prefix updates occur once per complete inner row, then once per +complete inner rectangle, and so on. With side lengths at least two their +number is less than twice the record count. Singleton axes are folded into +one constant per rectangle. Rewinding a factor page costs its length and is +charged to the corresponding completed row. Ordinary O(p)-bit multiplication +therefore gives O(V polylog p) scan cost with O(gp) temporary bits. This +argument uses bounded normalized **prefix products**; bounds on the final +product alone do not suffice. + +It does **not** cover an unbounded collection of different short kernels +inside one tile, a product-tree level with factor-two padding on all g axes, +or an unspecified transform-dependent sparse correction. Those require a +new representation theorem. In particular, the Gaussian_fast/product-tree +algorithm is not silently substituted for the translation-invariant kernel +assumed here. + +## Finite evidence + +`check_halo.py` verifies 5,632 exact outputs in three two/three-axis cyclic +examples, with asymmetric signed kernels, nontrivial diagonal conjugations, +periodic coarse carries, scrambled coarse order, unique sparse gather/scatter +keys and all local sequential crops. Pure flattening gives 3,040 wrong +boundary outputs across these examples; all are repaired. The exact copied +volumes equal `3V[(1+2L/N)^g-1]` in every case. These are constructive finite +controls; the preceding scan, metadata and volume arguments give the +conditional asymptotic claim. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/MANIFEST.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/MANIFEST.json new file mode 100644 index 0000000000..48dbb2a6bf --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/MANIFEST.json @@ -0,0 +1,20 @@ +{ + "EMBEDDING-CONTROLS.json": "d0182e172f16b55ec64a5edfabb10816e416348ef2db15449f3ed4b47f3fec1f", + "HALO-CONTROLS.json": "ae370f7d7421f2f96e101a4b9d8082110507c47211220fd6ead065c2366aa828", + "HALO-THEOREM.md": "cb9e1da38400b658f9209e465b4e5896fadd17dc66afa862f38bae8d8c1f2058", + "PARAMETER-CHECK.json": "e38cdfc13f1959186d888383458c9030c4bd3c8c3915aea4743d55637d2f8d76", + "RAGGED-CONTROLS.json": "c3688f65565486b585727374abfbe478f7544aa61381bd73cc6ffa168b4221b6", + "RAGGED-PERIODS.md": "c6e4dfe7f58dac7c5a513ffb3a9ab4801ab0717efae5c1e764e570335402ff25", + "README.md": "575a3acfc18a0e05ffe04a8563318a2770ce32cd7bf3a5e6123ce2de39a1cd4c", + "RECTANGULAR-EMBEDDING.md": "e64447bff7e7717802ee049290faaf29e45db3e36a6f7fcec109184478f15f26", + "SPARSE-SHEAR-CONTROLS.json": "d3b9763f7ef6a3a5752c6c11a8a4d5f465646ad75b178a1c1b3363540fcc4871", + "SUPPORT-ROUTER.md": "6ddb322e11883c1177ba6cd1a77d23a1dac827d7290b32eaf642457b1d762a54", + "check_embedding.py": "597cb5304498456575c99d46a7fb503e04db28d5a7f57857b102b50f703b7464", + "check_halo.py": "0a3257cbb8f37b6cc9bb44019cb114ea2ae9b28738a416bff4bc712d1306e1b7", + "check_ragged.py": "54b761cfe3404d524af3a2abdf09bcea8dfb5cca6199c0c182cee310e3ca8c78", + "check_sparse_shear.py": "6e22c989e6bae92c4cd209c9336aad4359956cf370bdbd5f6b70609144598121", + "embedding.log": "d0182e172f16b55ec64a5edfabb10816e416348ef2db15449f3ed4b47f3fec1f", + "halo.log": "ae370f7d7421f2f96e101a4b9d8082110507c47211220fd6ead065c2366aa828", + "ragged.log": "c3688f65565486b585727374abfbe478f7544aa61381bd73cc6ffa168b4221b6", + "sparse-shear.log": "d3b9763f7ef6a3a5752c6c11a8a4d5f465646ad75b178a1c1b3363540fcc4871" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/PARAMETER-CHECK.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/PARAMETER-CHECK.json new file mode 100644 index 0000000000..5844ead5c0 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/PARAMETER-CHECK.json @@ -0,0 +1,10 @@ +{ + "epsilon": "0.99833013302245016130167750009820221155623107175809", + "g_exponent": "0.00166985697754983869832249990179778844376892824191", + "K0_exponent": "0.00333972395509967739664499980359557688753785648382", + "halo_sort_exponent": "-0.4949904190673504839050325002946066346686932152743", + "halo_local_axes_exponent": "-0.4966602860449003226033550001964044231124621435162", + "tile_gauge_exponent": "0.50333972395509967739664499980359557688753785648381", + "K0_squared_exponent": "0.00667944791019935479328999960719115377507571296764", + "initial_gather_exponent": "0.0016669624023127060364416759538459168913455129064535" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-CONTROLS.json new file mode 100644 index 0000000000..553ea1a36a --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-CONTROLS.json @@ -0,0 +1,36 @@ +{ + "status": "PASS_RAGGED_TRUE_PERIOD_SPARSE_HALO", + "cases": [ + { + "S": [ + 19, + 22 + ], + "N": 8, + "L": 1, + "g": 2, + "input_records": 418, + "halo_records": 732, + "full_halo_candidate_pairs": 700, + "corrected_outputs": 274, + "source_and_rectangle_gathers_identical": true, + "true_period_tensor_convolution_checked": true + }, + { + "S": [ + 17, + 18, + 20 + ], + "N": 8, + "L": 1, + "g": 3, + "input_records": 6120, + "halo_records": 18064, + "full_halo_candidate_pairs": 14352, + "corrected_outputs": 4392, + "source_and_rectangle_gathers_identical": true, + "true_period_tensor_convolution_checked": true + } + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-PERIODS.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-PERIODS.md new file mode 100644 index 0000000000..fdfe26d197 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-PERIODS.md @@ -0,0 +1,89 @@ +# True periods inside a near-one dyadic container + +The primary halo theorem extends to arbitrary true periods s_i when an +already supplied dyadic rectangle t_i>=s_i has bounded total inflation +`product t_i / product s_i=O(1)`. Simultaneous zero embedding and its inverse +cost one ordered scan each by `RECTANGULAR-EMBEDDING.md`. This prerequisite +is satisfied only where the inherited near-one capacity bound applies; it +must not be replaced by independently rounding every axis to its next power. + +Take N dividing every t_i. Write `s_i=q_i N+r_i`, `0<=r_i0, one ragged core in axis i. The initial +support-limited gather uses the dyadic storage coordinates, not mixed-radix +division. It may produce invalid all-zero storage tiles; discard those. + +Apply ordinary flat cyclic convolution only to tiles whose every active +side has valid length N. Their interior outputs remain exact. Regular tile +face corrections use the earlier first-boundary-axis rectangles, but every +halo coordinate is now reduced modulo **s_i**, never t_i. + +If any side of a tile is ragged, discard its entire flat-convolution output. +For its actual side lengths r_i(tile), compute every valid output in +`product [0,r_i(tile))` using the input rectangle +`product [-L,r_i(tile)+L)`. All sides are at most N+2L. The same g successive +polynomial-length convolution/crop passes therefore produce exact outputs. +Full ragged tiles and regular face outputs form a disjoint correction set. + +Let Q_i=q_i+1_(r_i>0). The total full-halo volume of *all* valid tiles is + +``` +H_all = product_i [s_i + 2L Q_i]. +``` + +Their all-full-tile contribution is `product_i q_i(N+2L)`. The sum of the +ragged-tile halo volumes is their difference. Expanding by a first ragged +axis, or simply telescoping the product, gives + +``` +H_ragged <= H_all * sum_i (N+2L)/s_i + = O(S gN/s_min), +``` + +since `H_all/S <= exp(2L sum_i Q_i/s_i)=O(1)` when gL/N=o(1). +Regular face input volume is O(S gL/N). Thus total sparse copy volume is + +``` +M=O(V [gL/N + gN/s_min]). +``` + +The new term is smaller than V divided by every fixed polynomial in p when +`s_min=2^Theta(p/d)` and `N,g=poly(p)`. It pays all the same key sorting, +metadata and local computation factors. No new per-axis full scan appears. + +## One source-driven enumerator + +For each valid source coordinate x_i, list the valid destination tiles t +for which an unwrapped representative + +``` +y_i = x_i - tN + k s_i +``` + +lies in `[-L,r_i(t)+L)`. Only the home and neighboring core intervals can +qualify; there are at most three choices because L5/6, +all sparse halo work is O(V polylog p). The initial gather costs +`O(V g^(1-a_bit) polylog p)` and the grouped convolution costs +`O(V g^(1-q) polylog p)`. The cited bounded-gauge variation is still o(p). +Actual parameter exponents are recorded in `PARAMETER-CHECK.json`. + +Read: + +1. `SUPPORT-ROUTER.md`: arbitrary-spacing guarded shears, support-sized + reservoirs, fixed-tape repair, and the support-at-most-2k gather. +2. `HALO-THEOREM.md`: tensor identity, exact sparse halo volume, all record + movement, key lengths, local transposes, final output order and costs. +3. `RECTANGULAR-EMBEDDING.md`: simultaneous near-one zero embedding and crop + in one scan, using B=Omega(p) record width and an O(p)-bit descriptor. +4. `RAGGED-PERIODS.md`: arbitrary true periods inside that dyadic container, + using full ragged tiles as exponentially sparse additional exceptions. + +No rank/select oracle or sort of the complete tensor by a computed key is +used. Full scans preserve stream order. Only sparse correction records +are sorted, with every key bit pass charged. + +Reproduce, from the chat workspace: + +``` +python3 -B work/halo_layout_ultra/check_sparse_shear.py +python3 -B work/halo_layout_ultra/check_halo.py +python3 -B work/halo_layout_ultra/check_embedding.py +python3 -B work/halo_layout_ultra/check_ragged.py +``` + +Controls include an exhaustive 32,768-address shear permutation, 4,000 +unequal-spacing probes, 5,632 regular-grid convolution outputs and 6,538 +true-period inputs for ragged gather/correction checks. Wrong naive cyclic +flattening is explicitly detected before correction. Source/result hashes +are in `MANIFEST.json`. + +This is not a complete Gaussian resampler or a larger multiplication +exponent certificate. The kernel must be translation invariant and +separable; a fixed/polylogarithmic composition with bounded and sequentially +generated diagonal conjugations is admitted. A product tree with many +different short kernels or per-axis factor-two padding is not admitted by +renaming it a convolution. The missing Gaussian representation/normalization +theorem remains separate. The original bit/complex supplier, fixed-tape, +precision and all-size interfaces remain conditional inputs. + +Prepared with substantial OpenAI Codex assistance, continuing the retained +arbitrary-source router and bulk-resampling proofs in +`work/integer-mult-current/references/semantic-bulk/rad20/reports/`. +No frozen source or selected certificate was changed. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RECTANGULAR-EMBEDDING.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RECTANGULAR-EMBEDDING.md new file mode 100644 index 0000000000..191867c72e --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RECTANGULAR-EMBEDDING.md @@ -0,0 +1,51 @@ +# One-pass near-one rectangular embedding and crop + +This supplies the dyadic-container qualification for the halo theorem. It +does not require one full payload scan per axis. + +Let input axes have arbitrary lengths `s_1,...,s_D` and let dyadic target +lengths `t_i>=s_i` be given, with total address width +`A=sum_i log2 t_i=O(p)`. The input and output use lexicographic axis-major +order, the last coordinate changing fastest. Every payload record has at +least `c p` bits, as in the retained coefficient representation. Assume +`T=product t_i <= C product s_i = C S`; C is the separately proved +near-one capacity bound, not a product of per-axis dyadic roundings. + +Enumerate the complete target rectangle in lexicographic order with its +ordinary A-bit binary counter. For each target address x, output the next +source record if `x_i infinity`; short remaining cases can use the retained +full-span router, since `n=O(k polylog p)` there. The conditional cost is + +``` +O(V (k+1)^(1-a) polylog p + poly(p)). +``` + +All other coordinate positions return to their original values. This is a +support bound, not a bound on how far a moved coordinate travels. + +## Unequally spaced target lemma + +Put `B=2^G`, `G=4 ceil(log2 p)+6`, and `K=64G`. Take target positions +`j_0<...2(k+1)ell`, a gap outside the moved support contains a contiguous ell-bit +field S disjoint from the actual last ell bits. Use an original equal-ell +interchange A to move S to that actual suffix. Route the conjugate +`A pi A^-1` on the n-ell prefix only, then undo A. The conjugate fixes the +suffix pointwise because pi fixed S. Choose every dirty control reservoir +inside the prefix after this interchange, then recompute positions and +residue colors. Thus no control operation touches the enlarged record. +Its 2^ell complete values absorb the polynomial descriptor work; no data +is padded or duplicated. If `n<=2(k+1)ell`, the retained full-span bound +already gives `O(V (k+1)^(1-a) polylog p)`. The placement itself costs only +O(V polylog p). This explicit suffix-conjugation bookkeeping was +independently supplied in `work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md`. + +## The gather uses only S union T + +Let S list the k selected low-coordinate bits in the desired tensor order, +and T list k consecutive destination slots. Send S bijectively to T. Send +the remaining domain slots `T\S` bijectively to the remaining output slots +`S\T`, and fix everything outside `S union T`. This explicitly extends the +requested gather to a coordinate permutation supported on at most 2k bits. + +The other coarse-coordinate bits can be scrambled by this extension. Their +order is retained and described by the inverse permutation; **no stable +compaction of all coarse fields is performed**. A coarse neighbor address +can be decoded, incremented in its logical axis, then encoded in this +physical coarse order when constructing a sparse correction key. + +## Evidence and boundary + +`check_sparse_shear.py` checks all 32,768 addresses for targets 0 and 6 with +guard length 5, including 512 good addresses, inverse composition and every +bad repair. It also checks 4,000 arbitrary/constructed-good addresses with +unequal gaps 9 and 11 and sources inside guard fields. `check_halo.py` +checks three explicit support-at-most-2k gather permutations. + +These controls support the algebra; the cost argument above is the +all-size conditional transfer. It retains the original bit-supplier, +equal-width interchange, fixed-tape and common-label hypotheses. It does +not upgrade an arbitrary full computed-key sort to a fast router. + +Sources: the retained reports `compact-arbitrary-source-routing.md` and +`review-arbitrary-routing.md` under +`work/integer-mult-current/references/semantic-bulk/rad20/reports/`. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/check_embedding.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/check_embedding.py new file mode 100644 index 0000000000..54556ee93d --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/check_embedding.py @@ -0,0 +1,22 @@ +"""Exact simultaneous zero embedding/cropping in lexicographic order.""" +from itertools import product +from pathlib import Path +import json,math +HERE=Path(__file__).resolve().parent +def check(s,t): + original=list(product(*(range(n) for n in s)));it=iter(enumerate(original,1)) + embedded=[];valid=[] + for x in product(*(range(n) for n in t)): + if all(a=N-2*L] + for j in active: + for side in (0,1): + choices=[] + for i,a in enumerate(x): + if i=N-L:opts.append((1,a-N)) + else: + if a>=N-2*L:opts.append((0,a)) + if a=N-L:opts.append((1,a-N)) + choices.append(opts) + for selected in product(*choices): + dest=tuple((tile[i]+selected[i][0])%Q[i] for i in range(g)) + y=tuple(v[1] for v in selected) + result.append(((j,side,dest,y),value)) + return result + +def radix(records,key): + pairs=[(key(k),k,v) for k,v in records] + bits=max((k.bit_length() for k,_,_ in pairs),default=0) + for b in range(bits): + zero=[];one=[] + for row in pairs:(one if row[0]>>b&1 else zero).append(row) + pairs=zero+one + assert [k for k,_,_ in pairs]==sorted(k for k,_,_ in pairs) + return [(k,v) for _,k,v in pairs],bits + +def key_encoder(N,L,g,Q): + # Deliberately scramble coarse address significance. This represents the + # displaced coordinate slots in a support-limited gather. + coarse=[(i,j) for i,q in enumerate(Q) for j in range(q.bit_length()-1)] + coarse=coarse[::2]+coarse[1::2][::-1] + def enc(key): + j,side,tile,y=key + out=j*2+side + for i,b in coarse:out=(out<<1)|((tile[i]>>b)&1) + for a in y:out=(out*(N+2*L))+(a+L) + return out + return enc + +def replay(N,L,g,Q): + assert N>4*L and N&(N-1)==0 and all(q>=2 and q&(q-1)==0 for q in Q) + S=[N*q for q in Q];global_ranges=[(0,s) for s in S] + raw={x:1+sum((i+3)*a*a+(i+1)*a for i,a in enumerate(x)) for x in box(global_ranges)} + pre=lambda x:1+sum((i+1)*a for i,a in enumerate(x))%3 + post=lambda x:1+sum((i+2)*a for i,a in enumerate(x))%2 + data={x:v*pre(x) for x,v in raw.items()} + kernels=[[((i+2)*(z+L+1))*(-1 if z==0 else 1) for z in range(-L,L+1)] for i in range(g)] + shifts=list(product(range(-L,L+1),repeat=g)) + weight={z:math.prod(kernels[i][a+L] for i,a in enumerate(z)) for z in shifts} + direct={x:post(x)*sum(weight[z]*data[tuple((a-b)%s for a,b,s in zip(x,z,S))] for z in shifts) for x in data} + merged={};wrong=interior=0 + for tile in product(*(range(q) for q in Q)): + core=[data[tuple(t*N+a for t,a in zip(tile,unflat(u,N,g)))] for u in range(N**g)] + for u in range(N**g): + x=unflat(u,N,g);glob=tuple(t*N+a for t,a in zip(tile,x)) + val=post(glob)*sum(w*core[(u-sum(z[i]*N**(g-1-i) for i in range(g)))%(N**g)] for z,w in weight.items()) + wrong+=val!=direct[glob] + if all(L<=a0 + tiles=math.prod(Q) + expected_copies=tiles*2*sum((N**j)*(3*L)*(N+2*L)**(g-j-1) for j in range(g)) + assert len(copies)==expected_copies + expected_boundary=tiles*(N**g-(N-2*L)**g) + assert boundary==expected_boundary + return dict(N=N,L=L,g=g,coarse_sizes=Q,input_records=len(data),interior_outputs=interior, + corrected_boundary_outputs=boundary,gathered_halo_records=len(copies),predicted_halo_records=expected_copies, + direct_flattening_wrong_outputs=wrong,radix_key_bits=keybits, + exact_tensor_outputs_checked=len(data),diagonal_conjugations_checked=True, + scalar_coefficients='signed asymmetric integer product kernel') + +def main(): + cases=[replay(8,1,2,[2,4]),replay(8,1,3,[2,2,2]),replay(16,2,2,[2,2])] + perms=[gather_permutation([5,7,6],3),gather_permutation([4,4,4],3),gather_permutation([9,6],4)] + out=dict(status='PASS_EXACT_CORE_PLUS_SPARSE_BOUNDARY_TENSOR_CONVOLUTION',cases=cases, + support_limited_gathers=perms, + scope='Finite exact map, sparse-copy completeness, unique scatter, radix ordering and support controls; asymptotic fixed-tape costs are in the companion proof.') + (HERE/'HALO-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) +if __name__=='__main__':main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/check_ragged.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/check_ragged.py new file mode 100644 index 0000000000..4665063452 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/check_ragged.py @@ -0,0 +1,69 @@ +"""Independent source-driven versus rectangle-driven ragged halo gather.""" +from itertools import product +from collections import defaultdict +from pathlib import Path +import json,math +HERE=Path(__file__).resolve().parent +def points(ranges):return product(*(range(a,b) for a,b in ranges)) +def rectangles(S,N,L): + Q=[(s+N-1)//N for s in S];g=len(S) + for tile in product(*(range(q) for q in Q)): + sizes=[min(N,s-t*N) for s,t in zip(S,tile)] + if min(sizes)>(i*G))&((1<>(j+1))&((1<<(K-1))-1))<(1<<(K-1))-2*B + for i,j in enumerate(targets)) + +def ops(targets,sources,G): + out=[] + F=[('y1',1),('t1',1),('y2',1),('t2',-1)] + return F+[('load',1)]+F+[('load',-1)] + +def run(state,targets,sources,G,n,inverse=False): + u,t,y=state;H=len(targets)*G;mask=(1<>j)&1)<<(i*G) for i,j in enumerate(targets)))&mask + elif tag=='t2':t=(t+sign*sum((((y>>j)&1)^(digit(u,i,G)&1))<<(i*G) for i,j in enumerate(targets)))&mask + else:u=(u+sign*sum(((y>>s)&1)<<(i*G) for i,s in enumerate(sources)))&mask + return u,t,y + +def ideal(state,targets,sources): + u,t,y=state + return u,t,y^sum(((y>>s)&1)<=K for a,b in zip(sorted(targets),sorted(targets)[1:])) + assert all(j+K<=n for j in targets) + count=good_count=bad_count=corrected=0;images=set() + for state in states: + count+=1;out=run(state,targets,sources,G,n) + assert run(out,targets,sources,G,n,True)==state + assert good(*out,targets,G,K)==good(*state,targets,G,K) + assert good(*ideal(state,targets,sources),targets,G,K)==good(*state,targets,G,K) + if good(*state,targets,G,K): + good_count+=1;assert out==ideal(state,targets,sources) + else: + bad_count+=1 + fixed=ideal(run(out,targets,sources,G,n,True),targets,sources) + assert fixed==ideal(state,targets,sources) + corrected+=fixed!=out + images.add(out) + return dict(targets=targets,sources=sources,G=G,K=K,n=n,states=count, + good=good_count,bad=bad_count,changed_by_repair=corrected,distinct_images=len(images)) + +def main(): + small=check_case([0,6],[4,8],1,5,11, + product(range(4),range(4),range(1<<11))) + assert small['states']==small['distinct_images']==32768 + rng=random.Random(713) + targets=[1,10,21];sources=[5,15,25];G=2;K=7;n=29 + arbitrary=[(rng.randrange(64),rng.randrange(64),rng.randrange(1< infinity`. Their aggregate and all helper reservoirs + fit the full address because beta<1. There is no necessary constraint + beta<1-epsilon; a group is allowed to be wider than one original axis. +* The width-dependent phase sums are geometric. With q=2`, the exponent is at least +`h^2-|h|`; the two-sided tail is at most `4 exp(-2a)`. Thus the claimed +`exp(-2a theta)+5 exp(-2a)` bound follows. In particular, counting both nearest +neighbors separately at the larger bound would be invalid, but the supplied +argument does not do so. + +The actual H is specified by globally fixed rounded weights and a globally +fixed truncated Toeplitz kernel inside each physical phase cell. Its diagonal +is one, with the omitted aliases included in the perturbation. The candidate's +perturbation budget is strictly below the displayed row-dominance gap. Principal +submatrices preserve this row gap. Adding identity hole rows/columns preserves +it as well, and increases the physical bandwidth by at most the q-image spacing +factor, which is bounded by two. + +For `F=H_aug-I`, `||F||_infinity<=1-delta` with +`delta>=1/(16p)`. Powers through order J agree on a core row whose distance from +the principal-window edge exceeds J times the bandwidth. The two inverse tails +are bounded by `2(1-delta)^J/delta`. With `J=C p^2`, this is exponentially small +in p. The distance requirement is `O(p^(5/2) polylog p)`, below the candidate's +`A=C_A p^3` eventually. The preserved physical period cut and the fact that +true axes exceed every polynomial in p exclude spurious short wrap edges. + +## 2. Broad Gaussian conditioning is a different argument, and it works + +Sources: candidate lines 151–201; +`work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md` lines 13–73 and 75–105; +`work/gaussian_fast_ultra/BILATERAL-LDU.md` lines 25–119. + +For the finite LDU formulas, I checked the origin-shift correction and the +triangular inverse estimates. If `h=j-i`, the elementary-polynomial estimate +for the upper inverse reduces its exponential part to + +`b[beta_j^2-beta_i^2-rho^2 h-2 rho sum_{k=i}^{j-1} beta_k]`. + +Using `|beta_k|<=1/2` gives at most `b/4-b rho theta h`. The lower inverse uses +the corresponding expression with the other sign on the beta sum and has the +same decay. The factors `c_b^-1` and `c_b^-2` come from the explicitly bounded +products. This proves the stated uniform finite-interval bound; integrality of +the q-values is not needed, only gaps at least one and the beta bound. + +The broad target-cell identity also checks directly. With +`q'_k=k+rho c`, `beta'_k=theta k-rho c`, and +`b=b0/rho^2`, one obtains + +`K=alpha^-1 N_b diag(exp(-b beta'^2))`. + +At a selected target index, `beta'_k` equals the original source beta. Therefore +the finite LDU bound really applies to the block K used in the candidate; it is +not a bound for a different narrow matrix. The product estimate +`log(c_b^-1)<=pi^2/(12 b rho)` gives +`c_b^-4<=exp(pi alpha^2 rho/3)=p^O(1)`. The reciprocal gap is also polynomial +because `theta^-1=poly(p)` and `alpha^2=O(log p)`. + +The square identity-hole completion has block form `[A 0; R I]`. Its inverse is +`[A^-1 0; -R A^-1 I]`. Hence bounded Gaussian row sums and the selected-row +inverse bound give a polynomial full inverse bound. Dividing the completion +by four is contractive even on numerical hole errors. Its extra factor of +one half per axis is exactly compensated by the stated `2^D` shift. + +For the Toeplitz core, the Poisson symbol bound supplies a polynomial lower +eigenvalue, since its exponent is `O(alpha^2)=O(log p)`. An exponentially small +truncation/rounding perturbation preserves that bound. The extension from +these SPD cores to shallow circuits is outside this subreview, but the +Gaussian side supplies the conditioning claimed for that extension. + +## 3. Capping changes the runtime gauge, without changing H + +Sources: candidate lines 203–221 and 273–283; +`work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md` lines 31–107; +`work/phase_assembly_ultra/CAPPED-INTERIORS.md` lines 13–29 and 68–70. + +Within a fixed narrow phase cell, + +`log(a_i/a_j)=a(i-j)(beta_i+beta_j)`. + +Its absolute value is at most `a|i-j|`. Taking exact ratios of the already +rounded positive weights at one local anchor cancels the same common scalar +from both sides of the similarity, so the matrix is unchanged. Relative +rounding error of the original weights remains small because their minimum +is `exp(-O(ad))` and the absolute setup precision is `Cp`, with `ad=o(p)`. +Thus the local circuit sees range `O(aB)`, not the original `O(ad)`. + +For the broad split, expansion of the displayed exponents gives exactly +`-b0(delta+x-lambda y)^2`. The two logarithmic derivatives are +`-2 b0 lambda beta'_i` and `2 b0 lambda^2 beta'_k`; each is bounded in absolute +value by b0. Changing anchor changes the two ideal diagonals by reciprocal +constants. Applying the same exact reciprocal change to previously rounded +diagonals preserves their product with the Toeplitz core. Thus the broad +rounding scheme can consistently use the same global matrix and locally +recenter its capped restrictions. + +The number of pieces does not multiply the norm of their direct sum. The +group logarithmic range is consequently `O(gaB+g polylog p)`. The inherited +absolute weight accuracy adds `O(ad)` once to the precision requirement, rather +than once for every tensor axis. The candidate's condition +`beta+xi+r0=1-zeta`, with `a=O(log p)<=p^r0` eventually, makes this range o(p). +The total precision is therefore `p+O(ad+gaB+g polylog p)=O(p)` in both parameter +families. This conclusion concerns asymptotics; no reasonable finite cutoff +is supplied or required by the statement. + +## 4. Boundary factorization and rounding do not introduce a window-count factor + +Sources: candidate lines 225–240; +`work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md` lines 78–98; +`work/integer-mult-current/references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md` +lines 203–301. + +The five-factor identities check by ordinary block multiplication. The shears +use B and C from the original banded matrix. They do not use the generally +dense products `A^-1 B` or `C A^-1`. For forward expansion, F is only applied; +its inverse is not required by that formula. + +For a strictly row-diagonally-dominant matrix, one elimination step gives + +`gap_i(new)>=gap_i(old)+|a_ik/a_kk| gap_k(old)`. + +Its absolute row norm does not increase. Thus the narrow Schur matrix retains +the polynomial gap and bounded row norm. A noncyclic cut of its banded matrix +also retains row dominance. In its LU factors, U has bounded row norm and the +unit-lower factor has only O(W) entries per row, each bounded by the reciprocal +pivot gap. Therefore `||L||=poly(p)`. The identities +`L^-1=U A^-1` and `U^-1=A^-1 L` then give polynomial inverse-factor norms. +The periodic correction has rank O(W) and is controlled by the full completed +inverse, as in the retained identity `K^-1=I-V A^-1 U`. + +The exact partial right-hand sides are consequently polynomially bounded. +Rounding a completed block update creates a row residual bounded by +`poly(p) 2^-P`. Equivalently, rounding and coefficient error can be treated as +a perturbation whose norm is small enough to close the same bound on computed +values. Applying the polynomial full inverse bounds the output error by that +residual. Infinity norm takes the maximum over windows rather than summing +their count. Strassen's fixed recursion contributes polynomial W growth and +O(log W) guard bits, already covered by the precision allowance. + +## 5. Density, stream layout, and global calls + +Sources: candidate lines 207–211, 242–271, and 368–385; +`work/phase_layout_audit_ultra/AUDIT.md` lines 15–188; +`work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md` lines 7–65; +`work/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md` lines 16–58; +`work/halo_layout_ultra/HALO-THEOREM.md` lines 88–196. + +The capped carve contributes O(d/B+log B) separators per physical phase cell. +Retaining W-wide separators ensures distinct interiors are uncoupled. Each +elimination only joins adjacent O(W)-sized boundary groups, so the Schur +bandwidth remains O(W). The extra clipped-endpoint term in an L-core is +O(W polylog p), negligible relative to `L W/B polylog p`. + +For the union U, every sparse factor changes its own axis only inside E_i; +therefore it preserves U. The dense A stage does not preserve U, and the +candidate explicitly scatters and regathers around it. For one sparse axis, +`U_i=E_i x [L]^(g-1)` is a complete rectangle inside U. Extracting it in order, +transposing only its polynomial-width axis field, processing, undoing the +transpose, and merging back needs no full g-axis key sort. The full union +scan is charged once per sparse axis, producing the stated g-squared factor +per group. The matrix corollary cannot afford an extra full-key union sort, +but this algorithm does not require one. + +The q-image predicate checks the exact half-open interval +`s(2k-1)<=2tj 2(k+1)L. +``` + +Indeed, after excluding the actual suffix, the at most k moved positions +cut the remaining interval into at most k+1 gaps. If every gap had length +less than L, that prefix would have length at most (k+1)L-1. The displayed +inequality contradicts this bound. + +Let A be the old disjoint equal-L swap exchanging S with the actual suffix. +Execute A, route the conjugated permutation A pi A^{-1}, then execute +A^{-1}. Because pi fixes S pointwise, its conjugate fixes the actual suffix +pointwise. It still moves at most k coordinates. All endpoints are now in +the n-L prefix. Choose the three dirty reservoirs in that prefix and run +all compact rotations, original control shuttles, elementary CNOTs, and +bad-hole repair there. Recompute endpoint names after A and after the +reservoir placement. In the application, n-L>=9kG holds with a wide +asymptotic margin. + +The suffix is a complete binary field with arbitrary contents. Reinterpreting +its 2^L values as part of a record gives record length R0*2^L, without +creating, padding, or deleting data. This length exceeds every fixed +polynomial in p. No operation whose cost relies on that record length acts +on its component coordinates. Both instances of A use the already approved +primitive and cost O(V L^(1-a))=O(V polylog p); they do not invoke the new +router. This removes the prospective circularity. + +For the general router outside this sparse branch, the retained +first-L/second-L/actual-last-L candidate convention and matching cohorts +also work. Arbitrary candidates should not be substituted blindly: a +candidate can partially overlap the actual suffix, whereas the old swap +requires equal disjoint fields. After each candidate placement, the same +n-L prefix exclusion is mandatory. If the entire binary span is only +polylogarithmic, the old elementary construction suffices. If +n=O((k+1)polylog p), the old full-span construction gives the desired +support bound up to polylogarithms; its own short-record preparation must +still use the old suffix construction. + +For a full H supplier call, reserve the suffix outside its selected field +and keep it fixed throughout the completed call. If necessary, perform +the same old L-bit placement before exposure, update the logical names of +the neighborhoods, and restore it after inverse exposure. The source +neighborhoods need not remain physically consecutive before exposure: +the exposure permutation explicitly routes their named bits. The target +complete chunks are placed before the reserved suffix. + +## 3. Whole-row padding and the next diagonal scan + +The leading row prefix and the front/back controls are assembled from +complete K-bit neighborhoods in the globally complete binary tensor. +Their bits are not assumed zero or semantically independent of coefficient +values. Completeness means every coordinate value is represented in the +stream. Multiplication by previous coarse-dependent diagonal coefficients +does not remove any address or change this property. + +Deferring the selected bits lying in reserved chunks is valid: the accepted +active invocation is the desired tensor on the other selected bits and +identity on every reserved coordinate, for arbitrary incoming coefficients. +The preliminary individual transforms in the old layout were used to +finish those selected bits, not to create complete row or control ranges. +After deferral, those transforms are performed in later phases instead. + +The row count is rounded once to a multiple of the actual product + +``` +Q_rows = W_complex^D_complex * W_bit^D_bit. +``` + +The padded row label is allowed to be a nonbinary preceding fiber count. +Every within-row field remains a full binary field. The supplier's +interchanges and controls act within those complete fields; the global +exposure router is not applied to the padded row label. Nested bit adapters +consume and restore the bit stock before later complex splits, exactly as +in the retained product-stock argument. + +At the completed invocation boundary, the operator is identity on the row +label. A padded zero row therefore returns to zero even though temporary +role computations may have combined or moved its contents. Delete those +rows before inverse exposure or the next deferred phase. This bounds each +phase's volume by twice its original volume, not by a factor exponential in +the number of phases. + +All H calls in the compiler have the same selected-bit schedule for every +coarse label; label dependence resides in the intervening diagonal tables. +Run one H call on the complete global tensor, complete every deferred +phase, remove padding, and restore the original exposed-coordinate order. +Only then run the next diagonal scan. The microbox layout used for that +scan is the restored layout in which the local coordinates are already +contiguous (or the separately paid low-bit gather layout). No padded row, +partial H result, or temporarily relocated coarse label is presented to +the diagonal-table lookup. Thus scanning different coarse-label tables +does not require separate H invocations or borrowing fixed labels inside +a microbox. + +## 4. The actual parameter inequalities + +The largest exposure moves k=O(gK)=O(p^.093 log p) binary coordinates. +With G=Theta(log p), its three dirty reservoirs require + +``` +9kG = O(p^.093 log^2 p) = o(p). +``` + +The suffix occupies only L=Theta(log^2 p) positions. In particular, + +``` +n/(kG) = Omega(p^.907/log^2 p) -> infinity, +n/((k+1)L) = Omega(p^.907/log^3 p) -> infinity. +``` + +There is ample room both for the old spectator placement and for all +subsequent prefix-only reservoirs. These inequalities concern the full +binary address, not any one original axis. + +The only per-axis inequality is + +``` +K + O(log p) < ell_axis = Theta(p^.002), +``` + +which holds eventually. The original K/log p -> infinity wording must be +replaced, as already noted in `SUPPLIER-SUPPORT-AUDIT.md`, by the finite +guard inequality K>=G+4ceil(log2 p)+10. K=128G satisfies it. + +The supplier's front/back controls are made by concatenating reserved +complete neighborhoods. Their chunk count is + +``` +ceil(2hG/K) + ceil(hG/K) <= 3h/128 + 2. +``` + +The leading row stock adds O(log h) complete chunks. Above a fixed +polylogarithmic cutoff this is at most h/16, and smaller h is handled +individually. No single original axis stores hG control bits. Indeed +gG/ell_axis=Theta(p^.091 log p) diverges; the construction works precisely +because it does not demand this false per-axis fit. + +Consequently the reviewed mechanisms support the proposed +O(V g^(1-q) polylog p) global-call cost under the retained supplier +assumptions. The suffix exclusion and completed-call boundary should be +kept explicit in any final assembly. They are necessary bookkeeping, +not a new exponent restriction or an unproved sorting primitive. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER-CONTROLS.json new file mode 100644 index 0000000000..80cb2c8083 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER-CONTROLS.json @@ -0,0 +1,154 @@ +{ + "status": "pass", + "arithmetic": "exact symbolic rational/complex", + "routing": [ + { + "N": 2, + "permutations": 2, + "switch_depth": 1 + }, + { + "N": 4, + "permutations": 24, + "switch_depth": 3 + }, + { + "N": 8, + "permutations": 40320, + "switch_depth": 5 + }, + { + "N": 16, + "random_permutations": 100, + "switch_depth": 7 + }, + { + "N": 32, + "random_permutations": 100, + "switch_depth": 9 + }, + { + "N": 64, + "random_permutations": 100, + "switch_depth": 11 + } + ], + "fixed_H_word_checks": [ + { + "kind": "upper", + "parameter": "-10", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "-10", + "H_calls": 6 + }, + { + "kind": "upper", + "parameter": "-3/7", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "-3/7", + "H_calls": 6 + }, + { + "kind": "upper", + "parameter": "0", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "0", + "H_calls": 6 + }, + { + "kind": "upper", + "parameter": "7/9", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "7/9", + "H_calls": 6 + }, + { + "kind": "upper", + "parameter": "10", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "10", + "H_calls": 6 + }, + { + "kind": "diag", + "H_calls": 6 + }, + { + "kind": "swap", + "H_calls": 6 + }, + { + "kind": "fft", + "H_calls": 6 + }, + { + "kind": "diag", + "H_calls": 6 + } + ], + "coefficient_odometer": [ + { + "N": 2, + "g": 1, + "records": 2, + "coefficient_products": 2, + "products_per_record": 1.0, + "table_movement_bound": 2, + "zeros_in_table": true + }, + { + "N": 2, + "g": 8, + "records": 256, + "coefficient_products": 510, + "products_per_record": 1.9921875, + "table_movement_bound": 510, + "zeros_in_table": true + }, + { + "N": 4, + "g": 5, + "records": 1024, + "coefficient_products": 1364, + "products_per_record": 1.33203125, + "table_movement_bound": 1364, + "zeros_in_table": true + }, + { + "N": 8, + "g": 4, + "records": 4096, + "coefficient_products": 4680, + "products_per_record": 1.142578125, + "table_movement_bound": 4680, + "zeros_in_table": true + } + ], + "tensor_test": { + "N": 4, + "axes": 2, + "data_coordinates": 16, + "input_output_matrix_entries": 256, + "simultaneous_H_calls": 18, + "expected_H_calls": 18, + "dimension_expansion": false, + "different_axis_matchings": true, + "different_axis_gate_kinds": true, + "local_matching_endpoints_checked": true + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER.md new file mode 100644 index 0000000000..19efb9cb1c --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER.md @@ -0,0 +1,160 @@ +# Tensorizing local circuits without dimension padding + +Status: an algebraic compiler and scan-cost lemma. It reduces a tensor product of different bounded-depth local circuits to a bounded sequence of simultaneous binary Hadamard calls and diagonal scans on the **same** data volume. The retained supplier must support the stated Hadamard call on selected local bits and restore the agreed endpoint record order. No new fast-transform supplier is asserted here. + +## 1. Interface and conclusion + +Let N=2^m and let g axes each have N local coordinates. A coarse-label block contains V=N^g data records in lexicographic local-coordinate order. On axis a let C_a be an N-wire linear circuit of depth D; every layer consists of disjoint invertible two-coordinate gates and one-coordinate diagonals. The permitted gates are bounded shears, swaps, FFT butterflies, and diagonals. Circuits, pair matchings, coefficients and depths can differ by axis and by coarse label. Their tables are fixed independently of data values. + +Write H=[[1,1],[1,-1]]. A simultaneous Hadamard call means the tensor product of one H acting on one selected bit in each axis, with all remaining local and coarse coordinates as spectators. It acts on existing 2^g-coordinate fibers inside the N^g records; it allocates no new coordinate. The bit selected in an axis may be prescribed by the compiler. + +After padding shorter circuits by identity **operations**, the tensor product C_1 tensor ... tensor C_g is a sequence of + + at most (8m+2)D simultaneous H calls + +and O(Dm) tensor-diagonal scans. A tensor diagonal has the form + + d(i_1,...,i_g;c)=product_(a=1)^g d_a(i_a;c), + +where c is the fixed coarse label. Each diagonal scan uses O(1) amortized coefficient multiplications and O(1) amortized coefficient-table record movements per data record, after loading gN local values. The number of arithmetic-data coordinates remains exactly N^g throughout. + +If all local gate coefficients and inverses are bounded by B and all original prefix and inverse-prefix norms by Lambda, the compiler's local prefix bounds are Lambda times poly(B), up to absolute constants. For polynomial B and Lambda this is N^O(1). The tensor-prefix logarithmic bound is O(g log N). More generally depth D=polylog(p), B=poly(p), N=poly(p) gives an O(g polylog(p)) guard even without a stronger prefix theorem. + +An outer similarity whose logarithmic range is Q_a on axis a adds only sum_a Q_a. Therefore the retained phase-cell case Q_a=O(a d) uses + + p + O(g a d + g polylog(p)) + +working bits. The condition g a d+g polylog(p)=O(p) is sufficient for O(p)-bit records. + +## 2. Fixed Hadamard words, including zero shears + +All formulas below are matrix-product order; execution reads right to left. Let U(t)=[[1,t],[0,1]]. Direct multiplication gives + + H diag(2,1) H diag(-3,1) H = [[-8,-10],[0,-6]], + diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H = U(5/4). + +For q nonzero, conjugate by A_q=diag(4q/5,1): + + U(q)=A_q diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H A_q^-1. + +This has three H factors. To include t=0 with uniformly invertible diagonals, choose a known B>=|t| and c=2B+1. Then + + U(t)=U(t+c) U(-c). + +Both parameters have magnitude at least B+1 and at most 3B+1. This is a **six-H word for every t**, with all scalar coefficients and inverses bounded by poly(B). There is no division by t and no case split at t=0. Transpose the word to obtain the same six-H count for lower shears. + +There are two bounded identity words: + + I = (1/2) H H, + I = diag((1-i)/4,(1+i)/4) H diag(1,i) H diag(1,i) H. + +The second identity uses three H factors. Its middle product before the leading diagonal equals diag(2+2i,2-2i). Complex coefficients are permitted by the retained complex H interface. + +Consequently every gate type can be given **exactly six H factors**, separated by diagonal matrices: + +- Upper/lower shear: the six-H word above. +- Diagonal, including I: append three two-H identity words. +- Swap J: use J=(1/2)H diag(1,-1)H, then append two two-H identity words. +- FFT butterfly H diag(1,w), optionally scaled by 1/sqrt(2): append a two-H identity and a three-H identity. Here |w|=1. + +Combining neighboring diagonal matrices puts every word in one common alternating shape with six H factors and seven diagonals. Thus different gate types, zero parameters and inactive pairs can coexist in one simultaneous schedule without branches or padded data coordinates. + +## 3. Every matching can be routed to adjacent pairs + +For one local layer, assign its oriented pairs (i,j) to positions (2k,2k+1). Unpaired coordinates are assigned arbitrarily to the remaining positions, in pairs carrying diagonal or identity gates. This defines a permutation pi on exactly N coordinates. The layer equals + + P_pi^-1 (direct sum of adjacent-pair gates) P_pi. + +We use a Beneš network of 2m-1 switch layers. At each layer all switches join indices differing in one fixed bit; the bit sequence is + + 0,1,...,m-1,m-2,...,1,0. + +Here is a constructive proof of rearrangeability. Form a bipartite multigraph whose left vertices are input pairs {2j,2j+1} and whose right vertices are output pairs {2k,2k+1}. Input i supplies an edge to output-pair floor(pi(i)/2). Every vertex has degree two, so each component is an even cycle (parallel edges are allowed). Alternately color each cycle's edges 0 and 1. The first switch sends an input edge into the subnetwork of its color; the last switch sends that color to the required output parity. Each color class induces a permutation on N/2 pair indices. Recursively route both color subnetworks in parallel. This gives exactly the displayed bit sequence and uses no storage coordinates. + +The switch is either I or J. Both are obtained from the common two-H word + + S_epsilon=(1/2) H diag(1,epsilon) H, + epsilon=+1 for I, epsilon=-1 for J. + +Thus all axis-dependent switch controls are diagonal entries. A physical switch on bit k simply uses a simultaneous H call selecting bit k in each axis. No interstage wire shuffle is needed. If a supplier accepts only a contiguous selected-bit field, the existing paid bit router must gather that field and undo the gather; that is a supplier-interface cost, not an uncharged permutation. + +Each forward or inverse Beneš route uses 2(2m-1) H calls. The adjacent gates use six. Hence one original matching layer uses + + 2[2(2m-1)]+6 = 8m+2 + +H calls. The schedule is identical on every axis and every coarse-label block; only diagonal tables change. + +## 4. Tensor fusion is literal multiplication, not a sum expansion + +Let the compiled axis-a circuit be D_(a,L) H_(a,L) ... H_(a,1) D_(a,0), with the same number and arrangement of H factors. Kronecker multiplication gives + + tensor_a C_a + = (tensor_a D_(a,L)) (tensor_a H_(a,L)) ... + (tensor_a H_(a,1)) (tensor_a D_(a,0)). + +Every tensor of diagonals is a single diagonal scan. Every tensor of binary H factors is one simultaneous H call. This introduces neither a sum of 2^g branches nor a 2^g padding factor. The 2^g-dimensional Hadamard fibers are subsets of the N^g existing coordinates. + +An axis with no work at a particular original depth receives an identity layer, routed and compiled with the same fixed shape. Its intermediate H operations are still present, but the complete local word is I. Coarse-label variation is equally harmless: for each fixed label the identity above holds separately, and the whole data operator is a direct sum over labels. + +## 5. Uniform intermediate bounds and precision + +A completed switch route is a permutation and has norm one. Each partial switch word has absolute constant norm and inverse norm. Between original local circuit boundaries, at most one original gate word is partially executed. Its length is constant, and its scalar coefficients and inverses are poly(B); all different pairs act as a direct sum. Therefore + + max_prefix ||C_compiled,prefix||, + max_prefix ||C_compiled,prefix^-1|| <= poly(B) Lambda. + +The length O(D log N) does not require multiplying this polynomial bound afresh at every stage: completed original prefixes already have bound Lambda; completed route and padding words have their exact intended endpoints. The corresponding suffix bound is also polynomial when the original endpoint and inverse-prefix bounds are. + +For g tensor factors, the operator norms multiply, so their logarithms add. A microbox coarse-label direct sum takes the maximum over labels, not a product over labels. Hence polynomial local bounds give exp(O(g log N)); using only the rough original bound Lambda<=(2B)^D gives exp(O(gD log B)). Adding the original cell similarity contributes exp(sum Q_a). + +The number of compiled stages is polylog(p). Allocate each rounding or supplied-transform error below the final tolerance divided by this stage count and the prefix/suffix amplification. This costs O(g polylog(p)+sum Q_a) guard bits. Supplier-internal precision charges still must be those already guaranteed for the simultaneous H call; this compiler does not erase them. + +## 6. Product diagonals with constant amortized scan cost + +Fix one diagonal stage and one coarse label. Load the gN coefficients d_a(i) on a coefficient tape in axis order. Use base-N lexicographic enumeration with axis 0 fastest. Store suffix products + + P_j=product_(a=j)^(g-1) d_a(i_a), 1<=j=1) k/N^k = O(V), + +plus O(g) initialization. No division by d_a(i) is used, so this method also permits zero diagonal entries when an external application permits them. Every cached product is recomputed from current factors, never updated by a long chain of multiplicative ratios, so numerical drift does not depend on the number of records scanned. + +The same accounting works on a fixed number of tapes. Keep the coefficient-tape head normally in axis-0's N-record table. Rewinding that table costs O(N) once per N records. A depth-k carry visits the needed tables up to axis k and returns, moving O(kN) coefficient records. Summing over carries gives + + O(V sum_(k>=1) kN/N^k)=O(V) + +record moves. The suffix-product tape is accessed near its front except on carries, with the same geometric bound. Counter/address arithmetic costs only polylog(p) per record. This does not require g work tapes or a random-access lookup every record. + +The table loading cost is O(gN) records per coarse block and stage. For N>=2 it is O(N^g), including small g, and polynomial-time table construction is negligible per N^g block in the retained asymptotic regime g=p^beta tending to infinity. If tables are supplied rather than constructed locally, no additional assumption is needed. Both generation and loading costs must be charged once per visited coarse block, not silently once for the entire input. + +At O(p) working precision, O(1) amortized scalar multiplications per record and O(1) record movements cost O(V_bits polylog(p)) under the retained pointwise-arithmetic kernel, where V_bits counts bit volume. There is no g V_bits coefficient-generation term. + +## 7. Exact controls + +`check_tensor_compiler.py` constructs the network controls and all fixed words. `TENSOR-COMPILER-CONTROLS.json` records: + +- every permutation for N=2,4,8, including all 40,320 permutations at N=8; +- 100 random permutations each at N=16,32,64; +- exact symbolic checks of upper/lower zero and nonzero shears, diagonal, Swap and FFT words; +- an exact two-axis N=4 replay with different axis matchings and gate types: 16 data coordinates, all 256 input-output matrix entries, 18 simultaneous H calls, no dimension expansion. + +The finite controls support the constructive proofs; the cycle coloring, word identities, Kronecker identity and geometric scan sums establish the asymptotic statements. + +## 8. Cost substitution into the retained supplier + +Suppose a simultaneous H call on g selected bits, across V_bits bits of data and spectators, has paid cost O(V_bits g^(1-q) polylog(p)). Suppose also that any required bit-field gather/restore has cost O(V_bits g^(1-a_bit) polylog(p)). For D=polylog(p), N=poly(p), the compiler uses only polylog(p) such calls, and its diagonal scans cost O(V_bits polylog(p)). Thus the grouped local circuit costs + + O(V_bits [g^(1-q)+g^(1-a_bit)+1] polylog(p)). + +When a_bit>q and 0= j else row[j-i] + for j in range(n)] for i in range(n)]) + + +def lower(c): + return toep(c) + + +def invseries(f): + n = len(f) + ans = np.zeros(n, complex) + ans[0] = 1 / f[0] + for j in range(1, n): + ans[j] = -sum(f[k] * ans[j-k] for k in range(1, j+1)) / f[0] + return ans + + +def conv(a, b, n): + return np.convolve(a, b)[:n] + + +def diag(indices, values): + return [[((int(i),), np.array([[v]], complex)) for i, v in zip(indices, values)]] + + +def inverse(plan): + return [[(ids, np.linalg.inv(g)) for ids, g in layer] + for layer in reversed(plan)] + + +def parallel(a, b): + return [(a[j] if j < len(a) else []) + (b[j] if j < len(b) else []) + for j in range(max(len(a), len(b)))] + + +def fft(indices): + n = len(indices) + if n <= 1: + return [] + assert n & (n - 1) == 0 + bits = n.bit_length() - 1 + rev = [int(f"{j:0{bits}b}"[::-1], 2) for j in range(n)] + plan = [[((indices[i], indices[j]), np.array([[0, 1], [1, 0]], complex)) + for i, j in enumerate(rev) if i < j]] + size = 2 + while size <= n: + layer = [] + for start in range(0, n, size): + for j in range(size // 2): + w = np.exp(-2j * np.pi * j / size) + layer.append(((indices[start+j], indices[start+j+size//2]), + np.array([[1, w], [1, -w]], complex) / math.sqrt(2))) + plan.append(layer) + size *= 2 + return plan + + +def circ_shear(target, source, kernel): + """target += circulant(kernel) source, using these wires only.""" + p = parallel(fft(target), fft(source)) + lams = np.fft.fft(kernel) + mid = [[((source[j], target[j]), np.array([[1, 0], [lams[j], 1]], complex)) + for j in range(len(source))]] + return p + mid + inverse(p) + + +def toeplitz_shear(target, source, T): + """Cyclic + negacyclic decomposition; source and target are disjoint.""" + n = len(source) + assert T.shape == (n, n) and len(target) == n + c, r = T[:, 0], T[0, :] + g = np.zeros(n, complex) + h = np.zeros(n, complex) + g[0] = h[0] = c[0] / 2 + for j in range(1, n): + g[j] = (c[j] + r[n-j]) / 2 + h[j] = (c[j] - r[n-j]) / 2 + gamma = np.exp(1j * np.pi / n) + d = gamma ** np.arange(n) + twists = parallel(diag(target, d), diag(source, d)) + return (circ_shear(target, source, g) + twists + + circ_shear(target, source, d * h) + inverse(twists)) + + +def triangle_plan(indices, f): + n = len(indices) + STATS["triangle_nodes"] += 1 + STATS["max_triangle_coefficient"] = max( + STATS["max_triangle_coefficient"], float(np.max(np.abs(f)))) + if n == 1: + return diag(indices, f) + assert n % 2 == 0 + m = n // 2 + ev, od = indices[::2], indices[1::2] + e, o = np.asarray(f[::2]), np.asarray(f[1::2]) + k = conv(o, invseries(e), m) + zk = np.r_[0, k[:-1]] + s = e - np.r_[0, conv(o, k, m)[:-1]] + right = toeplitz_shear(ev, od, lower(zk)) + middle = parallel(triangle_plan(ev, e), triangle_plan(od, s)) + left = toeplitz_shear(od, ev, lower(k)) + return right + middle + left + + +def lowdisp_shear(target, source, T): + n = len(source) + Z = np.eye(n, k=-1) + Delta = T - Z @ T @ Z.T + U, vals, Vh = np.linalg.svd(Delta) + cutoff = 1e-11 * max(1.0, vals[0]) + rank = int(np.sum(vals > cutoff)) + STATS["max_displacement_rank"] = max(STATS["max_displacement_rank"], rank) + STATS["generator_shears"] += 1 + result = [] + reconstructed = np.zeros_like(T) + for j in range(rank): + g = U[:, j] + h = vals[j] * Vh[j, :] + STATS["max_generator_entry"] = max( + STATS["max_generator_entry"], float(np.max(np.abs(g))), + float(np.max(np.abs(h)))) + reconstructed += lower(g) @ lower(h).T + gamma = max(1.0, 2 * float(np.sum(np.abs(h)))) + p = h / gamma + p[0] += 1 + # U(h)=J L(h) J. This is a genuinely invertible source transform. + transform = triangle_plan(source[::-1], p) + result += transform + result += toeplitz_shear(target, source, gamma * lower(g)) + result += inverse(transform) + result += toeplitz_shear(target, source, -gamma * lower(g)) + assert np.max(np.abs(T - reconstructed)) < 1e-9 + return result + + +def full_plan(indices, M): + n = len(indices) + STATS["full_nodes"] += 1 + if n == 1: + return diag(indices, [M[0, 0]]) + m = n // 2 + A, B, C, D = M[:m, :m], M[:m, m:], M[m:, :m], M[m:, m:] + K = np.linalg.solve(A, B) + L = np.linalg.solve(A.T, C.T).T + S = D - C @ K + right = lowdisp_shear(indices[:m], indices[m:], K) + mid = parallel(full_plan(indices[:m], A), full_plan(indices[m:], S)) + left = lowdisp_shear(indices[m:], indices[:m], L) + return right + mid + left + + +def replay(plan, n, track=True): + out = np.eye(n, dtype=complex) + peak = 1.0 + min_det = 1.0 + for layer in plan: + used = set() + for ids, g in layer: + assert not used.intersection(ids), "non-disjoint layer" + used.update(ids) + min_det = min(min_det, float(abs(np.linalg.det(g)))) + out[list(ids), :] = g @ out[list(ids), :] + if track: + peak = max(peak, float(np.linalg.norm(out, 2))) + STATS["min_gate_determinant"] = min(STATS["min_gate_determinant"], min_det) + return out, peak + + +def main(): + rng = np.random.default_rng(20261013) + rows = [] + for n in [2, 4, 8, 16]: + f = rng.normal(size=n) / (12 * n) + f[0] += 1 + p = triangle_plan(list(range(n)), f) + actual, peak = replay(p, n) + rows.append({"kind": "triangular", "n": n, "rounds": len(p), + "gates": sum(map(len, p)), + "max_error": float(np.max(np.abs(actual-lower(f)))), + "peak_prefix_norm_2": peak}) + + c = rng.normal(size=n) / (20 * n) + r = rng.normal(size=n) / (20 * n) + c[0] += 1 + H = toep(c, r) + p = full_plan(list(range(n)), H) + actual, peak = replay(p, n, track=n <= 8) + rows.append({"kind": "full_toeplitz", "n": n, "rounds": len(p), + "gates": sum(map(len, p)), + "max_error": float(np.max(np.abs(actual-H))), + "peak_prefix_norm_2": peak if n <= 8 else None, + "input_distance_from_I_2": float(np.linalg.norm(H-np.eye(n), 2))}) + # Destructive control: dropping the second subtraction computes the wrong map. + # Independent direct parity block algebra across a range of sizes. + identities = [] + for n in [2, 4, 8, 16, 32]: + f = rng.normal(size=n) / n + f[0] += 2 + m = n // 2 + e, o = f[::2], f[1::2] + k = conv(o, invseries(e), m) + s = e - np.r_[0, conv(o, k, m)[:-1]] + E, O, K = lower(e), lower(o), lower(k) + Z = np.eye(m, k=-1) + left = np.block([[np.eye(m), np.zeros((m,m))], [K, np.eye(m)]]) + diagblock = np.block([[E, np.zeros((m,m))], [np.zeros((m,m)), lower(s)]]) + right = np.block([[np.eye(m), Z@K], [np.zeros((m,m)), np.eye(m)]]) + order = list(range(0,n,2)) + list(range(1,n,2)) + exact = lower(f)[np.ix_(order,order)] + identities.append({"n": n, "error": float(np.max(np.abs(left@diagblock@right-exact)))}) + sylvester = [] + for b in [1, 2, 4, 8]: + f = np.array([1.0]) + g = np.array([1.0]) + for j in range(b): + f = np.convolve(f, [1.0, 256.0 ** (j-b+0.5)]) + g = np.convolve(g, [1.0, 256.0 ** (-j-0.5)]) + fp = np.r_[f, np.zeros(3*b+2-len(f))] + fi = invseries(fp) + def hc(k): + return sum(g[v]*fi[k+v] for v in range(b+1) if 0 <= k+v < len(fi)) + K = np.array([[hc(i-b-j) for j in range(b)] for i in range(b)]) + H = np.array([[hc(i-j) for j in range(b)] for i in range(b)]) + S = np.zeros((2*b, 2*b), complex) + for j in range(b): + for k in range(b+1): + if j+k < 2*b: + S[j+k, j] += f[k] + if 0 <= b+j-k < 2*b: + S[b+j-k, b+j] += g[k] + inds = list(range(2*b)) + p = toeplitz_shear(inds[:b], inds[b:], K) + p += full_plan(inds[b:], H) + p += triangle_plan(inds, np.r_[f, np.zeros(2*b-len(f))]) + actual, peak = replay(p, 2*b, track=b <= 4) + sylvester.append({"b": b, "total_wires": 2*b, "rounds": len(p), + "gates": sum(map(len, p)), + "max_error": float(np.max(np.abs(actual-S))), + "H_minus_I_norm_1": float(np.linalg.norm(H-np.eye(b), 1)), + "H_minus_I_norm_infinity": float(np.linalg.norm(H-np.eye(b), np.inf)), + "peak_prefix_norm_2": peak if b <= 4 else None}) + # Removing the compensating dirty subtraction must fail. + b = 4 + gi, hi = np.array([1, .02, .01, .005]), np.array([.1, .04, .02, .01]) + pi = hi.copy() + pi[0] += 1 + target = np.block([[np.eye(b), np.zeros((b,b))], [lower(gi)@lower(hi).T, np.eye(b)]]) + pp = triangle_plan(list(range(b))[::-1], pi) + correct = pp + toeplitz_shear(list(range(b, 2*b)), list(range(b)), lower(gi)) + inverse(pp) + broken_actual, _ = replay(correct, 2*b, track=False) + correct += toeplitz_shear(list(range(b, 2*b)), list(range(b)), -lower(gi)) + correct_actual, _ = replay(correct, 2*b, track=False) + omission = {"correct_error": float(np.max(np.abs(correct_actual-target))), + "omit_second_subtraction_error": float(np.max(np.abs(broken_actual-target)))} + result = {"seed": 20261013, "note": "Floating replay supports, does not replace, algebraic proof.", + "controls": rows, "polyphase_identity": identities, + "normalized_sylvester": sylvester, "omission_control": omission, "stats": STATS} + (ROOT / "CIRCUIT-REPLAY.json").write_text(json.dumps(result, indent=2)+"\n") + print(json.dumps(result, indent=2)) + + +if __name__ == "__main__": + main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py new file mode 100644 index 0000000000..4f0180c88f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py @@ -0,0 +1,157 @@ +"""Exact rational checks of parity LU and displacement rank closure.""" +from fractions import Fraction as F +from pathlib import Path +import json +import random + + +def zeros(n, m=None): + return [[F(0) for _ in range(n if m is None else m)] for _ in range(n)] + + +def eye(n): + a = zeros(n) + for i in range(n): + a[i][i] = F(1) + return a + + +def mm(a,b): + return [[sum(a[i][k]*b[k][j] for k in range(len(b))) + for j in range(len(b[0]))] for i in range(len(a))] + + +def add(a,b,sgn=1): + return [[x+sgn*y for x,y in zip(r,s)] for r,s in zip(a,b)] + + +def inv(a): + n=len(a) + q=[r[:] + s for r,s in zip(a,eye(n))] + for j in range(n): + k=next(k for k in range(j,n) if q[k][j]) + q[j],q[k]=q[k],q[j] + v=q[j][j] + q[j]=[x/v for x in q[j]] + for i in range(n): + if i!=j: + v=q[i][j] + q[i]=[x-v*y for x,y in zip(q[i],q[j])] + return [r[n:] for r in q] + + +def rank(a): + a=[r[:] for r in a] + k=0 + for j in range(len(a[0])): + p=next((i for i in range(k,len(a)) if a[i][j]),None) + if p is None: + continue + a[k],a[p]=a[p],a[k] + v=a[k][j] + a[k]=[x/v for x in a[k]] + for i in range(k+1,len(a)): + v=a[i][j] + a[i]=[x-v*y for x,y in zip(a[i],a[k])] + k+=1 + if k==len(a): + break + return k + + +def block(a,b,c,d): + return [r+s for r,s in zip(a,b)] + [r+s for r,s in zip(c,d)] + + +def lo(f): + return [[f[i-j] if i>=j else F(0) for j in range(len(f))] + for i in range(len(f))] + + +def shift(n): + a=zeros(n) + for i in range(1,n): + a[i][i-1]=F(1) + return a + + +def displacement(a): + z=shift(len(a)) + return add(mm(z,a),mm(a,z),-1) + + +def series_inv(f): + g=[1/f[0]] + for i in range(1,len(f)): + g.append(-sum(f[j]*g[i-j] for j in range(1,i+1))/f[0]) + return g + + +def conv(a,b,n): + return [sum(a[j]*b[i-j] for j in range(i+1) + if j=j else r[j-i] for j in range(n)] for i in range(n)] + walk(H) + receipt={"status":"pass","arithmetic":"exact rational", + "parity_instances":parity,"schur_nodes":trace} + Path(__file__).with_name('EXACT-ALGEBRA.json').write_text(json.dumps(receipt,indent=2)+'\n') + print(json.dumps({"status":"pass","parity_instances":len(parity), + "parity_entries":sum(x['matrix_entries_checked'] for x in parity), + "schur_nodes":len(trace)},indent=2)) + + +if __name__=='__main__': + main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/check_supplier_capacity.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/check_supplier_capacity.py new file mode 100644 index 0000000000..65b20b8b68 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/check_supplier_capacity.py @@ -0,0 +1,47 @@ +"""Exact capacity controls in the large-g, gG >> single-axis-ell regime.""" +from pathlib import Path +from fractions import Fraction +import hashlib +import json + + +def main(): + # p itself has one million binary exponent bits. Only short descriptions + # and modest exact powers are needed; no input tensor is materialized. + logp=1_000_000 + eps=Fraction(998,1000) + beta=Fraction(93,1000) + interior=Fraction(907,1000) + d_exp=int(eps*logp) + g_exp=int(beta*logp) + ell_exp=logp-d_exp + g=1<ell + assert reserved<=g//16 + assert (g*K*G).bit_length() 2` is a fixed rational number. There is a single, uniform algorithm, with a fixed finite description, with the following properties. + +* Its recursive block multiplication uses a fixed finite bilinear formula whose coefficients are computable algebraic complex numbers. +* Given two `W` by `W` complex dyadic matrices with component magnitudes at most `2^h`, it computes their product with maximum entrywise absolute error at most `2^{-p}` in + +\[ +O\!\left(W^\tau M(p+h+\log(W+1))\right) +\] + +bit operations. Here `M(t)` is a regular integer multiplication bound: `M(t) >= t`, `M(Ct)=O_C(M(t))` for fixed `C`, and precision doubling sums to `O(M(t))`. All constants can depend on the chosen fixed block formula. +* If the input encoding has `h = O(p)`, this is + +\[ +O\!\left(W^\tau M(p+\log(W+1))\right). +\] + +If also `log W = O(p)`, it is exactly `O(W^tau M(p))`, without a suppressed logarithmic factor. In particular this holds for `W = Theta(sqrt(p))`. + +The strict exponent slack is essential to this statement: an exponent infimum does not by itself supply a finite scheme attaining that infimum. If a source proves `omega_C <= 9/4`, one may take any fixed rational `tau` with `9/4 < tau < 5/2`, for example `tau = 19/8`. + +The proof below also gives an exponent strictly below a specified threshold from only the promise that `omega_C` is below that threshold. It does not give useful numerical constants or an efficient way to discover the block formula. + +## 1. A finite exact bilinear formula exists with exponent slack + +Write `R(b)` for the exact complex tensor rank of `b` by `b` matrix multiplication. Use the usual equivalent definition + +\[ +\omega_{\mathbb C}=\inf_{b\ge2}\log_b R(b). +\] + +Thus some fixed integers `b >= 2` and `r` satisfy + +\[ +R(b)\le r= 2`, `r >= 1` satisfying `r < b^tau`; +2. finite encoded algebraic tuples of the required length; +3. exact tests of all rank equations. + +For rational `tau=a/d`, the first comparison is the integer comparison `r^d < b^a`. The search halts under the stated exponent promise, by Sections 1 and 2. Its output supplies the single fixed block formula. This search may be extremely expensive, but its cost is an absolute constant independent of `W,p,h`, so including it does not change the asymptotic bound. The same result can be phrased by fixing the first successful tuple once and for all. + +For just the promise `omega_C < 5/2`, search using `r^2 < b^5`. Upon success choose a rational `sigma` with + +\[ +\max\{2,\log_b r\}<\sigma<5/2. +\] + +The resulting algorithm has exponent at most `sigma < 5/2`. Thus one need not know a numerical gap in advance to obtain a uniform algorithm below the threshold. A practical certificate or practical constant is a separate issue. + +## 4. Each fixed algebraic coefficient has an ordinary `O(M(q))` precision algorithm + +Every chosen coefficient has a fixed minimal polynomial and a fixed isolating region. In characteristic zero its minimal-polynomial root is simple. Fix a rational starting approximation in a sufficiently small Newton convergence neighborhood. Newton iteration with successively doubled precision computes the root to absolute error `2^{-q}` with `O(M(q))` bit operations. + +The polynomial degree, coefficient heights, root separation, initial precision, and bounds on the derivative and its reciprocal are all fixed constants here. Polynomial evaluation, complex division, and the finite number of operations per Newton step cost `O(M(t))` at precision `t`; the usual precision-doubling sum is `O(M(q))`. This is not a claim uniform in algebraic degree or height. + +There are only finitely many block coefficients. Compute their dyadic approximations once at the precision needed for the current multiplication, and reuse them at all recursive nodes. This costs `O(M(q))` total, with a formula-dependent constant, rather than a logarithmic factor per scalar operation. + +## 5. Recursive arithmetic work + +Pad the matrices with zeros to size `N=b^L`, where `W <= N < bW` (and handle `W=1` directly). At each node, form the fixed `r` input linear combinations, recursively multiply the corresponding submatrices, and form the fixed output linear combinations. + +The arithmetic operation count satisfies + +\[ +T(N)\le rT(N/b)+O(N^2). +\] + +Let `rho=log_b r`. If `rho>2`, this is `O(N^rho)`; if `rho=2`, it is `O(N^2 log N)`; and if `rho<2`, it is `O(N^2)`. Every case is `O(N^tau)` because `rho2`. In particular the possible equality case `r=b^2` creates no extra factor in the advertised bound: its logarithm is absorbed by the fixed exponent slack. + +One could instead tensor a fixed formula a fixed number of times and add a zero product to arrange `b^2 < r < b^tau`, obtaining an exact recursion exponent `rho` strictly between two and `tau`. This refinement is unnecessary for the advertised `O(W^tau)` conclusion. + +## 6. Fixed-point roundoff grows only polynomially in the dimension + +Use fixed-point complex arithmetic with `q` fractional bits, round after scalar products, and use dyadic block coefficients whose absolute errors are at most + +\[ +\epsilon=2^{-q}. +\] + +Complex arithmetic introduces only absolute constant factors. Additions may be done exactly on this fixed grid; allowing an additional absolute error `O(epsilon)` after an addition also satisfies the following bounds. + +For a multiplication node with remaining depth `d`, let `a,beta >= 1` bound the maximum absolute entries of its two inputs. There is a fixed constant `K`, depending only on the block formula, such that the computed result obeys + +\[ +\|\widehat T_d(A,B)-AB\|_{\max} +\le \epsilon E_d a\,\beta, +\qquad +E_d\le K E_{d-1}+K b^{d-1},\quad E_0\le K. +\] + +Here the symbol `b` in `b^{d-1}` is the fixed block dimension; `beta` is the second input bound. + +To verify the recurrence, every local input linear combination has fixed length and fixed coefficients. Its rounded value differs from its exact value by at most `c epsilon a` or `c epsilon beta`, and its size is at most `S a` or `S beta`, for fixed `c,S`. Applying the inductive child bound therefore contributes at most + +\[ +c\epsilon E_{d-1}a\beta. +\] + +Perturbing the two child inputs in their *exact* matrix product contributes at most + +\[ +c\epsilon b^{d-1}a\beta, +\] + +because each entry of that child product sums `b^{d-1}` scalar products. Finally, the output reconstruction has fixed length and fixed coefficients. Propagation through its exact coefficients is a fixed factor; coefficient approximation and reconstruction rounding contribute at most + +\[ +c\epsilon\bigl(b^{d-1}+\epsilon E_{d-1}+1\bigr)a\beta. +\] + +For `epsilon <= 1`, all these terms have the stated recurrence after enlarging `K`. Taking `K >= 2b` gives `E_d <= C K^d` for a fixed `C`. Thus at the root + +\[ +\|\widehat T_L(A,B)-AB\|_{\max} +\le C\,2^{-q}K^L a\beta +=2^{-q}N^{O(1)}a\beta. +\] + +For root input magnitude at most `2^h`, take `a=beta=max(1,2^h)` and choose + +\[ +q=p+2\max(h,0)+\lceil L\log_2K\rceil+O(1). +\] + +The final absolute error is then at most `2^{-p}`. + +There is also no hidden exponential-in-precision magnitude growth. Along a path, each preprocessing step multiplies a magnitude envelope by a fixed `S`. A node at depth `t` therefore receives inputs bounded by `S^t a` and `S^t beta`. Its computed output is bounded by its exact product size plus the proved error: + +\[ +\bigl(b^{L-t}+\epsilon C K^{L-t}\bigr)S^{2t}a\beta. +\] + +Local scalar products and partial sums obey the same bound up to fixed factors. All intermediate magnitudes are consequently `N^{O(1)} max(1,2^{2h})`. Only `O(h+log N)` integer bits are needed. The total working bit width is therefore `O(p+h+log N)`. + +The polynomial error estimate exploits the structure of a bilinear recursion: linear preprocessing, one variable-by-variable scalar product at a leaf, and linear reconstruction. Bounding an arbitrary multiplication by the largest intermediate magnitude at every level would give an unnecessarily pessimistic estimate and is not the argument used here. + +## 7. Bit complexity and scope + +Every scalar operation uses `O(p+h+log W)` bits and costs `O(M(p+h+log W))`. Multiplying by the `O(W^tau)` arithmetic operation count proves the statement. Padding, input conversion, output rounding, and storage of the fixed coefficient approximations fit the same bound. + +The result supplies absolute accuracy. It does not promise small relative error in entries canceled nearly to zero. It applies to arbitrary fixed finite complex bilinear formulas; large fixed coefficient heights or norms worsen only the constants and the fixed multiple of `log W` in the precision requirement. + +An exponent theorem alone therefore suffices for computable, stable approximate multiplication with any fixed strictly larger exponent. It does not justify using the exact infimum exponent without slack, treating a varying family of coefficients as fixed, or claiming a concrete efficient implementation without constructing its constants. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/MANIFEST.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/MANIFEST.json new file mode 100644 index 0000000000..5a943f0fd6 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/MANIFEST.json @@ -0,0 +1,35 @@ +{ + "status": "MATRIX_BOUNDARY_TRANSFER_SIGNED_OFF", + "scope": "Full 9/4 source audit plus effective fixed algebraic rank scheme and approximate p-bit transfer; no final outer ledger promotion.", + "target_exponent": "19/8", + "source_rank_exponent_bound": "9/4", + "poly_W_error_growth": true, + "computable_fixed_selection": true, + "files": [ + { + "name": "PROOF-AND-COVERAGE.md", + "bytes": 12130, + "sha256": "db7509d754bb7fc3381419e0fed2674f22a9d57d34e8420fd5afa3c688fde9aa" + }, + { + "name": "ALGEBRAIC-STABILITY-INDEPENDENT.md", + "bytes": 11743, + "sha256": "7873e7d9a99fdc9de3ee224b9925d3e277e9773d7f5f8c2d0f1c19b0e52bc87f" + }, + { + "name": "SOURCE-COVERAGE.json", + "bytes": 2839, + "sha256": "fdc6849f97999a6cf99cdaa50591d24f78d6fb0bc580315f205452e5906b35cc" + }, + { + "name": "SOURCE-CONSTRUCTIONS.json", + "bytes": 21018, + "sha256": "114c1d8fb70592455a80c73d8a1bc6bfc9aca4e7395a420b6331d5474b4d1917" + }, + { + "name": "check_source_constructions.py", + "bytes": 6388, + "sha256": "9d387098403c0355be8e93158df02bae5a435b2366bd28efdf0b127e470c8c1c" + } + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/PROOF-AND-COVERAGE.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/PROOF-AND-COVERAGE.md new file mode 100644 index 0000000000..1a98011b1f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/PROOF-AND-COVERAGE.md @@ -0,0 +1,91 @@ +# Matrix boundary transfer: full-source audit and interface signoff + +Status: the supplied 9/4 source supports the matrix boundary substitution, with strict exponent slack. This is a theorem-level, computable asymptotic transfer. It is not a numerical finite-size rank certificate and it does not remove any supplier or computational-interface assumptions in the outer integer-multiplication argument. + +The independent reader for this note read all 1,022 lines of +`/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/build/paper.tex` +in order, including the full character-existence appendix and all bibliography entries, and also read the README. The companion 2.258 paper is unnecessary for this transfer and is not claimed as read in this new audit. + +## 1. Source coverage + +| Source lines | Material read | Audit result | +|---|---|---| +| 1–36 | Preamble, title, abstract | Establishes complex arithmetic setting. | +| 37–141 | Introduction, theorem and proof route | Explicitly promises an asymptotic exponent, not a competitive finite size. | +| 142–203 | Tensor semiring, rank, characters, detecting lemma | Uses exact rank and mutual restriction. Rank need only be subadditive; no false additivity assumption. | +| 204–235 | Oriented dot-product exponents | Monotonic completely multiplicative functions give power laws; tensor product of the three orientations is square matrix multiplication. | +| 236–269 | Degenerations and interpolation | Fixed degeneration degree is handled before the tensor-power limit. No uniform-in-power degree assumption is made. | +| 270–349 | Finite shared-leg separation | Fourier relation is exact since the phase has absolute size below 5M. Surviving weights equal `(g-h)^2`; the retained auxiliary dot index is distinct from the direct-sum block index. | +| 350–391 | Shared-leg entropy | The factor M cancels correctly before the N-th-root limit. Fixed-frequency counts and continuity cover every probability vector. | +| 392–432 | Polynomial multiplication and symmetrized profile | Rank is exactly a+b−1 by interpolation and output flattening. All six leg permutations are needed for the later middle branch. | +| 433–523 | Determinant filtration and discrete concavity | Fixed input/output bases work for every first-input slice. Multiplication preserves the determinant submodule; the forbidden upper-right block vanishes. The residual lower-left block receives positive weight. | +| 524–604 | Three sectors and shifted tripling | Outer sectors equal C(a,h); the middle is obtained by reversing the first basis and exchanging its second/third legs. This is an identification of an individual block and does not apply incompatible maps to the shared first leg. | +| 605–651 | Diagonal growth | Concavity gives the stated two-increment recurrence; H_a grows at least as a^(1/3). | +| 652–689 | Spectral bound, rank bound, recursion | Detecting characters may depend on d. The conclusion is `nu<=9/4`; selecting a fixed rank scheme requires a strict larger exponent. Rectangular block substitution preserves product order. | +| 690–699 | Algebraic-coefficient remark | The source explicitly allows replacing a complex solution of rational rank equations by a point over Qbar. | +| 700–769 | Appendix: invariant state family and multiplicativity | The fixed-point map is continuous, not asserted affine. Freezing its scalar eigenvalue produces a compact convex invariant subset, permitting finite intersections. | +| 770–905 | Appendix: detecting state existence | Finite rational separation gives an additive catalyst. Reuse keeps one s factor; no tensor cancellation is assumed. Amplification with a second multiplication tensor gives the contradiction to k9/4`. + +If the surrounding proof allows polylog p factors, the bandwidth's polylogs remain polylogs; no positive power of p is introduced by coefficient approximation or error amplification. The algorithm provides absolute accuracy, which is what local residual bounds use. It makes no relative-error promise for nearly canceled output entries. + +The recursion can also be scheduled with sequential tapes. At one fixed-base node, its fixed number of input block combinations and output block sums require a fixed number of complete scans and fixed block permutations of `O(n^2)` records. Store child inputs consecutively, process children depth-first, and stream their returned blocks into accumulators. Copying a pending-child region or repositioning a finite set of tapes costs another `O(n^2)`-record pass at that node, already covered by the recurrence's addition term. Addresses and recursion descriptions add at most logarithmic factors. If any implementation retains an additional `polylog W` factor, choose the rank scheme first with exponent strictly between 9/4 and the stated target tau, leaving enough fixed exponent slack to absorb it. No algebraic unit-cost random-access assumption is needed for the target power tau. + +## 4. Boundary substitution and the outer ledger + +The existing capped boundary algebra uses W simultaneous right-hand sides. A square product costs `O(W^tau M(p))`; dividing by W² processed entries gives `O(W^(tau-2) M(p))` per entry. One axis has `O(V W/B)` boundary entries, and there are d axes. Its total is consequently + +`O(V (d/B) W^(tau-1) [M(p)/p] polylog p)` + +when V is the total bit volume, as in the phase-assembly notes. There are V/Theta(p) scalar entries, so the factor M(p)/p is the per-bit arithmetic charge. Equivalently, if V counts scalar entries instead, the formula has M(p) without the division. The retained quasi-linear short multiplier has M(p)/p=polylog p. + +This retains the essential d/B factor. With `d=p^epsilon`, `B=p^b`, and `W=p^(1/2) polylog p`, its normalized exponent is + +`epsilon-b+(tau-1)/2`. + +Taking `tau=19/8` gives `(tau-1)/2=11/16` and boundary saving `5/16+b-epsilon`. This is already more than enough to put the basic sparse-union cost near the binding row at b near 3/4. Pushing tau down to the unattained endpoint 9/4 is unnecessary. + +The changed matrix algorithm has the same input/output interface as the former Strassen product. It does not change the original Gaussian matrix, boundary support, anchored gauge, square-block inverse choice, or number of right-hand sides. The surrounding local-residual proof must still supply polynomial bounds on local exact inverses and partial right-hand sides. The new product's error is `poly(p)2^-P`, so that proof applies with a fixed larger guard constant. + +The optional full-coordinate-key surcharge in the old arithmetic ledger would become binding if mechanically retained at b near 3/4. It is not part of the actual rectangle extraction/one-field transpose/ordered-merge algorithm, as confirmed by the separate phase-layout audit. Dropping it is a distinct layout fact, not a consequence of matrix multiplication. It must remain explicitly justified in the final outer ledger. All retained supplier assumptions remain in force. + +No final improved outer exponent is claimed in this note; that requires the whole ledger and the independent group-layout/gauge checks. + +## 5. Exact controls run in this audit + +`check_source_constructions.py` uses only Python's exact rational arithmetic and writes `SOURCE-CONSTRUCTIONS.json`: + +* Finite Fourier/separation support for M=1,...,12: the modular relation has no wraparound and zero weight is exactly g=h,u=v. +* Determinant change of basis for a=1,...,8 and b=2,...,8, for every first-input monomial: quotient and kernel diagonal slices are exactly the two claimed neighboring convolution tensors; the upper-right block vanishes. +* Three-sector identity for a,h=1,...,14: all weights are nonnegative and each of the three retained branches is exactly the claimed tensor after its permitted identification. +* Exact rational diagonal product lower bound through a=300. + +All controls passed. They detect several relevant implementation mistakes but are not a finite substitute for the universal arguments above or for the theorem's nonconstructive rank-existence step. + +## 6. Deliberate limitations + +There is no generated competitive finite matrix scheme, no numeric claim about its constants, no use of a rank formula at exponent exactly 9/4, and no fast algorithm for finding the first successful algebraic tuple. The algorithm remains computable and uniform because the finite search halts under the proved existence statement and its cost is independent of the growing input. The outer conclusion remains conditional on exactly the previously retained supplier/computational assumptions. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/SOURCE-CONSTRUCTIONS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/SOURCE-CONSTRUCTIONS.json new file mode 100644 index 0000000000..65880dc95c --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/SOURCE-CONSTRUCTIONS.json @@ -0,0 +1,1356 @@ +{ + "status": "PASS_EXACT_SOURCE_CONTROLS", + "scope": "Finite controls accompany a full universal proof audit; no rank algorithm synthesized.", + "source": "/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/build/paper.tex", + "source_sha256": "c396f2580a711b8b9e51ecf1265cb0e691acd745943f81bfc14ec0df209e1028", + "source_lines": 1022, + 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branches and the leg-swapped middle branch", + "arbitrary_arithmetic_circuit": "reject: stability proof requires recursive bilinear structure; unbounded sequential multiplication does not follow", + "endpoint_exponent_9/4": "reject: rank infimum only supplies fixed algorithms strictly above 9/4", + "arbitrary_complex_constants": "reject: computability requires algebraic-point replacement or explicit computable coefficients" + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/SOURCE-COVERAGE.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/SOURCE-COVERAGE.json new file mode 100644 index 0000000000..c6275bc1cb --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/SOURCE-COVERAGE.json @@ -0,0 +1,114 @@ +{ + "status": "FULL_SOURCE_READ_AND_AUDITED", + "source": "/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/build/paper.tex", + "sha256": "c396f2580a711b8b9e51ecf1265cb0e691acd745943f81bfc14ec0df209e1028", + "line_count": 1022, + "coverage": [ + { + "first_line": 1, + "last_line": 36, + "topic": "preamble and abstract", + "read": true + }, + { + "first_line": 37, + "last_line": 141, + "topic": "introduction and theorem", + "read": true + }, + { + "first_line": 142, + "last_line": 203, + "topic": "tensor semiring and rank", + "read": true + }, + { + "first_line": 204, + "last_line": 235, + "topic": "dot exponents", + "read": true + }, + { + "first_line": 236, + "last_line": 269, + "topic": "interpolation", + "read": true + }, + { + "first_line": 270, + "last_line": 349, + "topic": "finite shared-leg separation", + "read": true + }, + { + "first_line": 350, + "last_line": 391, + "topic": "entropy", + "read": true + }, + { + "first_line": 392, + "last_line": 432, + "topic": "polynomial profile", + "read": true + }, + { + "first_line": 433, + "last_line": 523, + "topic": "determinant filtration", + "read": true + }, + { + "first_line": 524, + "last_line": 604, + "topic": "tripling", + "read": true + }, + { + "first_line": 605, + "last_line": 651, + "topic": "diagonal growth", + "read": true + }, + { + "first_line": 652, + "last_line": 689, + "topic": "rank and arithmetic conclusion", + "read": true + }, + { + "first_line": 690, + "last_line": 699, + "topic": "algebraic coefficients", + "read": true + }, + { + "first_line": 700, + "last_line": 769, + "topic": "multiplicativity appendix", + "read": true + }, + { + "first_line": 770, + "last_line": 905, + "topic": "detecting-state existence appendix", + "read": true + }, + { + "first_line": 906, + "last_line": 1022, + "topic": "bibliography and ending", + "read": true + } + ], + "readme": { + "path": "/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/README.md", + "sha256": "1857fc06d04a43588207ba2f726679c50a418299a5fd0313423a354680a783a7", + "read": true + }, + "companion_2_258_claimed_read": false, + "companion_required": false, + "proof_dependencies": "The required new interpolation, determinant, entropy, growth and spectral detection arguments are contained in this source. Standard background dependencies are listed in PROOF-AND-COVERAGE.md.", + "finite_rank_scheme_generated": false, + "outer_improvement_claimed_by_this_audit": false +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/check_source_constructions.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/check_source_constructions.py new file mode 100644 index 0000000000..c8e362e90d --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/check_source_constructions.py @@ -0,0 +1,164 @@ +#!/usr/bin/env python3 +"""Exact finite controls for the constructions in the supplied 9/4 paper. + +These checks are controls for an end-to-end human proof audit, not substitutes +for the universal proofs or an executable fast matrix multiplication scheme. +""" +from fractions import Fraction as Q +from pathlib import Path +import hashlib +import json + +ROOT = Path(__file__).resolve().parent +SOURCE = Path('/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/build/paper.tex') + +def inverse(a): + n = len(a) + t = [[Q(v) for v in row] + [Q(i == j) for j in range(n)] + for i, row in enumerate(a)] + for j in range(n): + p = next(i for i in range(j, n) if t[i][j]) + t[j], t[p] = t[p], t[j] + z = t[j][j] + t[j] = [v/z for v in t[j]] + for i in range(n): + if i != j: + z = t[i][j] + t[i] = [v-z*w for v,w in zip(t[i],t[j])] + return [row[n:] for row in t] + +def mm(a,b): + return [[sum(x*y for x,y in zip(row,col)) for col in zip(*b)] + for row in a] + +def basis(e): + # Ambient coordinates: u^(e-i)v^i s, then u^(e-i)v^i w. + n = 2*(e+1) + cols = [] + for i in range(e+1): + v = [Q(0)]*n + v[i] = 1 + cols.append(v) + v = [Q(0)]*n + v[-1] = 1 + cols.append(v) + for j in range(e): + v = [Q(0)]*n + v[j+1] = -1 + v[e+1+j] = 1 + cols.append(v) + return [list(row) for row in zip(*cols)] + +def check_determinant(): + cases = [] + for a in range(1,9): + for b in range(2,9): + ei, eo = b-1, a+b-2 + bi, bo = basis(ei), basis(eo) + inv = inverse(bo) + for k in range(a): + mult = [[Q(0)]*(2*b) for _ in range(2*(a+b-1))] + for side in range(2): + for i in range(b): + mult[side*(a+b-1)+i+k][side*b+i] = 1 + t = mm(inv,mm(mult,bi)) + qo, qi = a+b, b+1 + for i,row in enumerate(t): + for j,val in enumerate(row): + if i < qo and j < qi: + assert val == Q(i == j+k) + elif i < qo and j >= qi: + assert val == 0 + elif i >= qo and j >= qi: + assert val == Q(i-qo == j-qi+k) + # Only the lower-left block may remain, with weight +1. + cases.append({'a':a,'b':b,'slices_checked':a}) + return cases + +def check_fourier(): + cases = [] + for m in range(1,13): + terms = surviving = zeros = 0 + for h in range(1,m+1): + for g in range(1,m+1): + for u in range(1,m+1): + for v in range(1,m+1): + terms += 1 + phase = u-v+2*(g-h) + assert abs(phase) < 5*m + if phase % (5*m) == 0: + surviving += 1 + assert phase == 0 + weight = g*g+h*u-h*h-h*v + assert weight == (g-h)**2 >= 0 + if weight == 0: + zeros += 1 + assert g == h and u == v + assert zeros == m*m + cases.append({'M':m,'potential_terms':terms, + 'fourier_survivors':surviving,'weight_zero_terms':zeros}) + return cases + +def check_tripling(): + cases = [] + for a in range(1,15): + for h in range(1,15): + B=3*h+a-1 + sectors=[set(),set(),set()] + for x in range(a): + for y in range(B): + z=x+y + sy=0 if y= 0 + if weight==0: + assert sy==sz + if sy==0: + t=(x,y,z) + elif sy==2: + t=(x,y-(2*h+a-1),z-(2*h+a-1)) + else: + # Swap second/third legs; reverse first-leg basis. + t=(a-1-x,z-(h+a-1),y-h) + assert t[2]==t[0]+t[1] + sectors[sy].add(t) + expected={(x,y,x+y) for x in range(a) for y in range(h)} + assert all(s==expected for s in sectors) + cases.append({'a':a,'h':h,'terms_in_each_branch':a*h}) + return cases + +def check_growth(): + h=Q(1) + for a in range(1,301): + if a>1: h *= 1+Q(1,3*(a-1)) + assert h**3 >= a + assert Q(3*a-1,2) >= a + return {'maximum_a':300,'exact_product_inequalities':'PASS'} + +if __name__ == '__main__': + raw=SOURCE.read_bytes() + result={ + 'status':'PASS_EXACT_SOURCE_CONTROLS', + 'scope':'Finite controls accompany a full universal proof audit; no rank algorithm synthesized.', + 'source':str(SOURCE), + 'source_sha256':hashlib.sha256(raw).hexdigest(), + 'source_lines':len(raw.splitlines()), + 'fourier':check_fourier(), + 'determinant_filtration':check_determinant(), + 'tripling':check_tripling(), + 'growth':check_growth(), + 'negative_controls':{ + 'drop_fourier_phase': 'reject: weights need not be nonnegative, e.g. M=2,h=2,g=1,u=1,v=2 has weight -5', + 'keep_only_one_outer_branch':'reject: factor 3 requires both outer branches and the leg-swapped middle branch', + 'arbitrary_arithmetic_circuit':'reject: stability proof requires recursive bilinear structure; unbounded sequential multiplication does not follow', + 'endpoint_exponent_9/4':'reject: rank infimum only supplies fixed algorithms strictly above 9/4', + 'arbitrary_complex_constants':'reject: computability requires algebraic-point replacement or explicit computable coefficients', + }, + } + (ROOT/'SOURCE-CONSTRUCTIONS.json').write_text(json.dumps(result,indent=2)+'\n') + print(result['status']) + print('source lines',result['source_lines']) + print('fourier M cases',len(result['fourier'])) + print('determinant cases',len(result['determinant_filtration'])) + print('tripling cases',len(result['tripling'])) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-INTERIORS.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-INTERIORS.md new file mode 100644 index 0000000000..62c3fa27b6 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-INTERIORS.md @@ -0,0 +1,72 @@ +# Conditional refinement: cap interior length independently of d + +This is a separate follow-up to the uncapped proof audit. Its current status is **PASS_CONDITIONAL_PHASE_FUSION_REVIEW**, following the independent capped-gauge and capped-layout audits and an integrated root review. It preserves the weaker `OUTER-ARITHMETIC.json` as history. Under the retained supplier assumptions and the reviewed interfaces, the capped construction below gives `kappa=0.001672922349588871`. The only new analytic change is the anchored O(aB) local gauge; all altered operation counts are derived explicitly here. No upstream selected certificate is changed. + +Let the original phase-cell scale remain `d=p^epsilon`, and cap dyadic interior lengths at `B=2^floor(b log2 p)`, where `1/20. The supplier support proof needs each individual K-neighborhood to fit its original axis and total moved support o(p); it does **not** require all g controls to fit within one original axis. The sufficient auxiliary inequality `beta<1-epsilon` in the uncapped ledger must therefore be removed in this follow-up. + +The changed principal exponents, normalized by V, are + +| Component | exponent | +|---|---:| +| main transform | `1-epsilon*q` | +| fused Gaussian circuits | `epsilon-q*beta` | +| boundary algebra | `epsilon-b+(omega-1)/2` | +| sparse union movement | `epsilon+beta+1/2-b` | +| grouped scans | `epsilon-beta` | + +For the boundary row, one axis has O(V W/B) boundary records. A band solve or coupling product on W simultaneous right-hand sides costs O(W^(omega-2)) per boundary record, so all d axes cost `O(V (d/B) W^(omega-1) polylog p)`. The new factor d/B must not be omitted. For the sparse union, its fraction within a group is O(gW/B polylog p); g sequential axis operations and d/g groups give `O(V (dg/B) W polylog p)`. These are precisely the third and fourth changed rows. + +All other costs in `OUTER-AUDIT.md` remain paid. Global routing has saving a_eff; packed polynomial products have saving epsilon. Even a conservative full active-group key surcharge adds only `beta+1-epsilon` to the union exponent. Sparse core-face corrections have exponent at most `epsilon+beta-3`, using the former p² per-record allowance on their O(gA/L) copied volume. + +With the rational bound omega<45/16, put c=(3-omega_upper)/2=3/32 and take a positive boundary slack s_boundary=10^-5. Balancing the two dominant rows while leaving that boundary slack gives + +``` +epsilon=(1+q+q*(c-s_boundary-r-zeta))/(1+q)^2, +b=epsilon*(1+q)-c+s_boundary, +beta=1-b-r-zeta, +kappa_limit=epsilon*q. +``` + +Using the same q, r and zeta as the uncapped candidate yields approximately + +``` +epsilon = 0.9984838711085557536432677624420451614... +b = 0.9064167934681446247757331487619919695... +beta = 0.0935821965318553752242668512380080305... +kappa_limit = 0.0016729223595888711324653863199468081... +kappa = 0.001672922349588871 +``` + +The boundary saving equals `kappa_limit+10^-5`. The union movement exponent is below 0.686, and even its optional full-key surcharge leaves it below 0.781. The per-record phase metadata exponent `beta+2(1-epsilon)` is below 0.097, so calculating all g membership predicates remains cheaper than scanning one p-bit record. Core-face repair costs remain negligible at L=p^8. + +The common compiled H calls are still made globally across complete coarse-label ranges. Each selected bit needs only its own K=128G neighborhood to fit its original axis. Since `1-epsilon>0`, this holds eventually even though gK is larger than one axis width. The total support is O(g polylog p)=o(p), because beta<1. All support-router reservoirs, product row stock and exact endpoint restoration are those of the independently audited supplier. + +At the displayed parameters, `b>1/2`, `epsilon>b`, `beta>0`, `epsilon>beta`, `1-b-r-beta=zeta>0`, and `1-epsilon-r>0`. These respectively pay the wide-band separation, cap below original cell scale, growing group payload, number of groups, compiled local-gauge precision, and global gamma/rounding range. The raw global rounded weights may still have O(ad) logarithmic range during offline generation. The **data** only sees exact anchored ratios of range O(aB); approximating those ratios to the retained precision introduces `exp(o(p))` error amplification, already paid by the strict group guard. + +The assembled group endpoints are the same full normalized Haug inverse or Saug/4 operators as in the uncapped proof. Changing the interior partition changes only their factorization. It does not change their completed contraction bounds, intended zero-hole semantics, principal-core locality, or the final `2^D` scale compensation. Consequently the uncapped precision and error argument applies with the improved group gauge budget. + +`check_capped_outer.py` computes 44 exact positive slacks, every displayed cost margin, and five rejecting controls. The full-cell-gauge control is especially relevant: it fails, so the stronger result depends on the independently reviewed anchored-gauge proof. The local review retains the original supplier and all-size conditional assumptions. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-OUTER-ARITHMETIC.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-OUTER-ARITHMETIC.json new file mode 100644 index 0000000000..56c871f4f3 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-OUTER-ARITHMETIC.json @@ -0,0 +1,388 @@ +{ + "status": "PASS_CONDITIONAL_PHASE_FUSION_REVIEW", + "scope": "Local conditional transfer and exact arithmetic; inherited supplier and all-size interfaces remain assumptions. 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b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY-ARITHMETIC.json new file mode 100644 index 0000000000..54ad0ed498 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY-ARITHMETIC.json @@ -0,0 +1,414 @@ +{ + "status": "PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW", + "scope": "Additional matrix-conditioned corollary. 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Fourths theorem gives omega_C<=9/4; apply the independently audited effective algebraic transfer at tau=19/8.", + "limitations": [ + "All finite supplier, common-frame, whole-call, semantic precision, fixed-tape and prime-packing assumptions of the first transfer remain.", + "No generated competitive finite matrix formula or practical runtime claim is made; the fixed formula search is effective but may have enormous constant cost.", + "This is a local conditional corollary, not an upstream selected certificate or an unconditional integer-multiplication theorem." + ], + "reviews": [ + { + "reviewer": "root agent", + "result": "Read matrix corollary and independent layout signoff; reproduced exact parameters/ledger and matrix construction controls; approved integrated conditional corollary." + }, + { + "reviewer": "matrix_boundary agent", + "result": "Read all 1022 supplied source lines and audited effective algebraic coefficient, fixed-tape, precision and rank-exponent transfer." + }, + { + "reviewer": "global_macro agent", + "result": "Independent new-parameter no-sort union, metadata, q-ordering and microhalo layout signoff." + }, + { + "reviewer": "phase_assembly agent", + "result": "Read both complete matrix transfer notes and independent layout signoff; constructed and replayed 47-slack ledger and six rejecting controls." + }, + { + "reviewer": "independent_fusion_review agent and independent children", + "result": "No blocking gap found under stated assumptions after full compiler, support, Gaussian, layout and tape-cost review. Generic SPD scale wording corrected without changing parameters. See independent FINAL-REVIEW.json." + } + ], + "arithmetic": { + "positive_slacks": 47, + "rejecting_controls": 6, + "minimum_saving": "0.0016733601154057784538170479517203762842018888767028492264912297200" + }, + "files": [ + { + "bundle_filename": "phase_fusion_note.tex", + "source_path": "work/phase_assembly_ultra/phase_fusion_note.tex", + "sha256": "08ecb1876695beee25484158e14cfd57c3d48f551a8a41d5abc998d71d674555", + "bytes": 33757 + }, + { + "bundle_filename": "MATRIX-COROLLARY.md", + "source_path": "work/phase_assembly_ultra/MATRIX-COROLLARY.md", + "sha256": "2d7fb1088e58bb86e574277733e082f135362215a69add71d9090213d32c9221", + "bytes": 5787 + }, + { + "bundle_filename": "check_matrix_corollary.py", + "source_path": "work/phase_assembly_ultra/check_matrix_corollary.py", + "sha256": "30db2d8e0f40fcc5c91e25f74cdc54e46aa4a0353c725345f9bcf5744e0b617f", + "bytes": 6692 + }, + { + "bundle_filename": "MATRIX-COROLLARY-ARITHMETIC.json", + "source_path": "work/phase_assembly_ultra/MATRIX-COROLLARY-ARITHMETIC.json", + "sha256": "c6efc4d2f6d17159cefebf5c2dd6099a6327fe0911c5dd87623cad96c7d6c56a", + "bytes": 16783 + }, + { + "bundle_filename": "PROOF-AND-COVERAGE.md", + "source_path": "work/matrix_boundary_ultra/PROOF-AND-COVERAGE.md", + "sha256": "db7509d754bb7fc3381419e0fed2674f22a9d57d34e8420fd5afa3c688fde9aa", + "bytes": 12130 + }, + { + "bundle_filename": "ALGEBRAIC-STABILITY-INDEPENDENT.md", + "source_path": "work/matrix_boundary_ultra/ALGEBRAIC-STABILITY-INDEPENDENT.md", + "sha256": "7873e7d9a99fdc9de3ee224b9925d3e277e9773d7f5f8c2d0f1c19b0e52bc87f", + "bytes": 11743 + }, + { + "bundle_filename": "MATRIX-SOURCE-COVERAGE.json", + "source_path": "work/matrix_boundary_ultra/SOURCE-COVERAGE.json", + "sha256": "fdc6849f97999a6cf99cdaa50591d24f78d6fb0bc580315f205452e5906b35cc", + "bytes": 2839 + }, + { + "bundle_filename": "MATRIX-SOURCE-CONSTRUCTIONS.json", + "source_path": "work/matrix_boundary_ultra/SOURCE-CONSTRUCTIONS.json", + "sha256": "114c1d8fb70592455a80c73d8a1bc6bfc9aca4e7395a420b6331d5474b4d1917", + "bytes": 21018 + }, + { + "bundle_filename": "MATRIX-TRANSFER-MANIFEST.json", + "source_path": "work/matrix_boundary_ultra/MANIFEST.json", + "sha256": "35d98d7579a16dae36530b4e5f4d0b8f925a069daa3f6fe1ea5276916167e0c2", + "bytes": 1159 + }, + { + "bundle_filename": "MATRIX-LAYOUT-SIGNOFF.md", + "source_path": "work/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md", + "sha256": "4c8c62f723232e842dc3bbdcc4defcc8ff1aeb86ba9f694661899dc6783b13c4", + "bytes": 3081 + }, + { + "bundle_filename": "MATRIX-LAYOUT-CONTROLS.json", + "source_path": "work/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json", + "sha256": "f8bd8729cb8438188e90fb9149f598b825461c97b4ed290584ca4367f812a48a", + "bytes": 1446 + }, + { + "bundle_filename": "PHASE-LAYOUT-AUDIT.md", + "source_path": "work/phase_layout_audit_ultra/AUDIT.md", + "sha256": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", + "bytes": 11774 + }, + { + "bundle_filename": "CAPPED-SIGNOFF.md", + "source_path": "work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md", + "sha256": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a", + "bytes": 4414 + }, + { + "bundle_filename": "CAPPED-GAUGE-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md", + "sha256": "1030a34922709ab8d3434a4c57a4a58934ae268c237fb598cb415697da411c9f", + "bytes": 9176 + }, + { + "bundle_filename": "LOG-A-PHASE-SCHUR-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md", + "sha256": "e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272", + "bytes": 10642 + }, + { + "bundle_filename": "FORWARD-SMALL-A-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md", + "sha256": "e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0", + "bytes": 6689 + }, + { + "bundle_filename": "SHALLOW-TOEPLITZ.md", + "source_path": "work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md", + "sha256": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310", + "bytes": 15480 + }, + { + "bundle_filename": "TENSOR-COMPILER.md", + "source_path": "work/inplace_algebra_ultra/TENSOR-COMPILER.md", + "sha256": "4d3b8466598a864a234b978e25976648d4e761840aad9153dd18d73d3dfe4d15", + "bytes": 13430 + }, + { + "bundle_filename": "SUPPLIER-SUPPORT-AUDIT.md", + "source_path": "work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md", + "sha256": "bd5b06eca043c5c871227264f6a954473c55cd16ec85bb420ff538fb8ec1d8d6", + "bytes": 16557 + }, + { + "bundle_filename": "SUPPORT-ROUTER.md", + "source_path": "work/halo_layout_ultra/SUPPORT-ROUTER.md", + "sha256": "6ddb322e11883c1177ba6cd1a77d23a1dac827d7290b32eaf642457b1d762a54", + "bytes": 7755 + } + ], + "post_review_correction": "Generic SPD prefix bound records both M and 1/m, not condition number alone; actual Gaussian bounds already satisfy it." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY.md new file mode 100644 index 0000000000..f08bb57f9c --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY.md @@ -0,0 +1,72 @@ +# Additional corollary using the supplied matrix-exponent theorem + +Status: **PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW**. The independent matrix-transfer and new-parameter layout audits passed, followed by the integrated root review and exact ledger replay. The first, Strassen-based result `kappa=0.001672922349588871` remains separately reviewed and unchanged. Neither result changes the upstream selected certificate. + +This corollary has an **additional hypothesis**: the supplied *Matrix Multiplication in Nine Fourths* theorem gives `omega_C<=9/4`. Its complete-source audit is `PROOF-AND-COVERAGE.md`; the effective approximate-bit transfer is `ALGEBRAIC-STABILITY-INDEPENDENT.md`. The result here retains every supplier and all-size assumption of the first phase-fusion theorem. + +## Effective matrix operation with the required error interface + +Fix `tau=19/8>9/4`. The matrix exponent theorem supplies a fixed block dimension m and rank r with `r0`. All local coefficient pages remain polynomial in p, while `L^g` remains superpolynomial. Microedge correction has exponent `epsilon+beta-3<-1.75`, including the generous old scalar arithmetic bound. Sparse microhalo key passes remain explicitly paid inside that negligible budget. + +All forward q-image group embeddings occur in native axis-major order before the one global low-bit gather. All compression q-image extractions occur after the inverse global gather. The entire Gaussian calculation keeps full dyadic target coordinate ranges, so every globally invoked H stage has complete coarse labels. This schedule needs no per-group gathering of the whole long-axis fields; the q-image scans have the already-paid Vd/g cost. + +The **optional full-group-key sort surcharge on the sparse union is omitted** because it is absent from the actual algorithm. The reviewed union routine extracts the complete rectangle `U_i=E_i x [L]^(g-1)`, transposes only its one polynomial-size field, computes its axis operator, and merges it into U in the original order. It does not radix-sort U. Reintroducing that sort is a rejecting control in the new ledger; it would destroy this parameter choice. This omission relies on the existing positive layout proof, not on the matrix theorem. + +`check_matrix_corollary.py` verifies 47 exact positive slacks and six rejecting controls and writes `MATRIX-COROLLARY-ARITHMETIC.json`. The rejected variants are endpoint equality, zero group guard, zero union slack, the old Vd row, the old full-cell data gauge, and adding a full-key sort to U. 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reservations and paid global synthetic layout", + "shallow Toeplitz plus SPD extension and tensor compiler", + "small-a forward conditioning and exact target-grid square completion", + "grouped sparse-union route and batched boundary arithmetic", + "dyadic core with sparse microbox-face repair and monotone q-image scans", + "scalar product-diagonal scan and retained all-size semantic/prime/error transfer" + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/OUTER-AUDIT.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/OUTER-AUDIT.md new file mode 100644 index 0000000000..974b8db67f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/OUTER-AUDIT.md @@ -0,0 +1,93 @@ +# Phase-cell fusion: outer assembly audit + +Status: assembly audit in progress. The exact arithmetic in this directory is a **conditional target**, not a replacement of the selected certificate. The remaining proof interfaces are recorded explicitly below. + +## Conditional transfer statement + +Assume the same finite bit/complex suppliers, common-label arithmetic compiler, semantic guard and all-size fixed-tape interfaces as the selected conditional PR1094 result. Also assume the new deferred-reservation, support-limited H supplier, shallow-circuit tensorization, and grouped sparse/core-halo lemmas below have their stated uniform implementations. Then the retained integer-multiplication reduction runs in + +`O(n (log n)^(1-kappa))`, `kappa=0.001672664764353193`. + +Here p=Theta(log n) and the live scalar bit volume V=Theta(n). Global one-axis preprocessing is polynomial in the original axis sizes and p, hence `n^o(1)`. Table construction or page loading repeated per coarse block is charged to its superpolynomial `L^g` payload by the tensor/scan lemmas; it is included in the displayed V costs, rather than called one global preprocessing operation. Every cost row below is `O(V p^e polylog p)` with `1-e>kappa`. The minimum excess is greater than `10^-11`. Consequently every fixed logarithmic factor and each use of the unconditional quasi-linear short arithmetic kernel can be absorbed in a single positive fraction of that excess. Floors, rounded alpha, and dyadic L alter constants or logarithmic factors. All claims are eventual in p; one fixed finite prefix of inputs can use the retained elementary fallback. This statement becomes an admitted theorem only when the named implementation hypotheses receive the intended full review. + +## Parameter convention + +Let `a_bit=0.001733427039589844234749` and `b=0.001675462676808103522925` be the retained lower bounds certified by the selected PR1094 replay. For applying the older compact recurrence, lower the usable bit saving to `a_eff=(b+q)/2`; this keeps its historical convention `a_eff0` with `(1-t_stop)b>a_eff` and `q<(1-t_stop)b`. + +For each visited coarse block and group, all required local coefficient pages can instead be regenerated in polynomial time in p: there are g local windows of size at most L=Theta(p^8), their rational origins have O(p/d) bits, and the published local coefficient/Schur/circuit constructors are polynomial in that size and requested precision. Recover local phases and anchor gauges from these origins; do not form a raw uncentered global q² exponential on the data. The page contains O(p)-bit online coefficients. Its polynomial setup and sequential loading are amortized by the complete `L^g` payload, which eventually dominates every fixed power of p. This gives an explicit access method even when the coarse labels are permuted; no random global coefficient-table lookup is assumed. + +Global chirps, twists, positive diagonal normalizations and exact scale factors are separable products. The prefix-product scan supplies their products in O(V polylog p), with completed scalar range `O(d alpha^2)=o(p)`. Packed polynomial products retain their saving epsilon. The finite supplier, exact common-label frame, whole-call hosting, semantic depth bound with C1=1, phase precision, prime packing and fixed-tape input/output assumptions are inherited unchanged. They are still conditional interfaces, as in the selected result. + +## Cost ledger, with conservative wide bandwidth + +Let `h_w=1/2` and `h_K=beta+1-epsilon`. Normalize all costs by V. Fixed logarithmic factors are absorbed by strict gaps. + +| Component | power of p | reason | +|---|---:|---| +| main transform | `1-epsilon*q` | deferred reservations | +| constant global coordinate routing | `1-a_eff` | paid support/full router | +| dense Gaussian groups | `epsilon-q*beta` | `(d/g) g^(1-q)` | +| grouped bit routing | `epsilon-a_eff*beta` | same grouping, bit supplier | +| one/fixed group scans | `epsilon-beta` | d/g groups | +| boundary algebra | `(omega-1)*h_w` | batched w-column Strassen | +| sparse union, basic movement | `beta+h_w` | V g w overall | +| sparse union, conservative key surcharge | `beta+h_w+h_K` | full active-group keys | +| polynomial products | `1-epsilon` | retained short polynomial products | +| core-face corrections | `epsilon+beta+3-8+2` | old local maps on sparse copies, with a loose p² allowance | + +Use the rational upper bound `omega<45/16`, checked by `7^16<2^45`. The target balances the first and third rows exactly before a fixed 10^-11 absorption backoff. + +This ledger is accepted only after its circuit tensorizer, grouped sparse route, target-grid core/halo adapter, forward small-a inverse bound, and deferred-reservation interfaces have written proofs. Exact positive inequalities alone do not discharge those interfaces. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/README.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/README.md new file mode 100644 index 0000000000..c9bd9bfd6f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/README.md @@ -0,0 +1,15 @@ +# Reviewed conditional phase-cell fusion + +With the explicit additional supplied matrix-exponent premise `omega_C<=9/4`, the reviewed local corollary gives **`kappa=0.001673360105405778`**, a gain of `6.9990832886e-7` over the selected conditional exponent. Its status is `PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW`. + +The separate Strassen-based transfer gives `kappa=0.001672922349588871` with status `PASS_CONDITIONAL_PHASE_FUSION_REVIEW`, without that additional matrix premise. Both retain the original supplier and all-size computational assumptions. The upstream selected certificate is unchanged. + +- `phase_fusion_note.tex` is the standalone proof note. +- `CAPPED-INTERIORS.md` records the local gauge and boundary-count change. +- `check_capped_outer.py` recomputes 44 positive rational slacks and five rejecting controls; `CAPPED-OUTER-ARITHMETIC.json` records the reviewed local status. +- `MATRIX-COROLLARY.md` and `check_matrix_corollary.py` give the additional corollary; `MATRIX-COROLLARY-ARITHMETIC.json` contains 47 positive slacks and six rejecting controls. `MATRIX-COROLLARY-REVIEW.json` records its separate review and source hashes. +- `OUTER-AUDIT.md`, `check_outer.py`, and `OUTER-ARITHMETIC.json` preserve the weaker uncapped arithmetic as history. + +Run `python3 -B work/phase_assembly_ultra/check_capped_outer.py` and `python3 -B work/phase_assembly_ultra/check_matrix_corollary.py` from the workspace root. + +The old `Vd` row and the unrecentered full-cell gauge both reject this result. The improvement uses the reviewed Gaussian circuits, support-limited compact supplier, anchored capped gauges, and grouped sparse/halo scans. Tightening the old arithmetic alone cannot produce it. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/REVIEW-RECEIPT.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/REVIEW-RECEIPT.json new file mode 100644 index 0000000000..2521d02ada --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/REVIEW-RECEIPT.json @@ -0,0 +1,53 @@ +{ + "status": "PASS_CONDITIONAL_PHASE_FUSION_REVIEW", + "created_utc": "2026-10-14T02:06:22.293042+00:00", + "kappa": "1672922349588871/1000000000000000000", + "kappa_decimal": "0.001672922349588871", + "upstream_selected_unchanged": true, + "scope": "Local all-size conditional transfer, retaining the selected finite suppliers and common-frame, whole-call, precision, fixed-tape and prime-packing hypotheses.", + "reviews": [ + { + "reviewer": "root agent", + "result": "Read standalone TeX, integrated notes and exact ledger; cap, supplier and layout audits passed; authorized local conditional status." + }, + { + "reviewer": "gaussian_fast agent", + "result": "Independent capped-gauge and finite-block audit passed; exact inherited-weight reanchoring and broad derivatives checked." + }, + { + "reviewer": "global_macro agent", + "result": "Independent capped carving, sparse-union, q-image and microedge layout audit passed." + }, + { + "reviewer": "inplace_algebra agent", + "result": "Global selected-H support and capacity audit passed; all g controls need not fit one original axis." + }, + { + "reviewer": "phase_assembly agent", + "result": "Read named proofs and relevant retained source contracts; reconstructed complete cost ledger and reviewed capped interfaces." + } + ], + "phase_assembly_replays": [ + "check_capped_outer.py: 44 positive exact slacks and 5 rejecting controls", + "check_forward_cell.py: 350 cases, 5184 factorization and 16496 completion entries", + "check_union.py: exact five-factor/tensor/complete-rectangle checks, with dense escape negative control", + "check_q_embedding.py: product order and augmented inverse checks" + ], + "files": { + "work/phase_assembly_ultra/phase_fusion_note.tex": "da2bf119be3ed1e44473c380e43643d16bdc93015c80380fc59da33d84e951de", + "work/phase_assembly_ultra/CAPPED-INTERIORS.md": "983c5cab2d2398f2192a8aec07daecb655f2de136e12a37f75672a14d27172ba", + "work/phase_assembly_ultra/check_capped_outer.py": "b4555c77de7a3ed4abc708a8d93363ef7c9a914be1091f80ab3ddea83f71230a", + "work/phase_assembly_ultra/CAPPED-OUTER-ARITHMETIC.json": "22caac9ec2ecf4d5743b7eb946b633911fbac55a714399556e87e2eae0457dec", + "work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310", + "work/inplace_algebra_ultra/TENSOR-COMPILER.md": "4d3b8466598a864a234b978e25976648d4e761840aad9153dd18d73d3dfe4d15", + "work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md": "bd5b06eca043c5c871227264f6a954473c55cd16ec85bb420ff538fb8ec1d8d6", + "work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md": "e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272", + "work/gaussian_fast_ultra/BILATERAL-LDU.md": "0d65e66d227464dda1a19f781ae0a0be671cbfd347c41e37ec954f3d629d8a77", + "work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md": "e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0", + "work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md": "1030a34922709ab8d3434a4c57a4a58934ae268c237fb598cb415697da411c9f", + "work/phase_layout_audit_ultra/AUDIT.md": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", + "work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a", + "work/halo_layout_ultra/SUPPORT-ROUTER.md": "d4d2f5bd0a8eb51d9ebd28e4d1a38d3c69bdbf97471a705b3aa199c184364c52", + "work/halo_layout_ultra/HALO-THEOREM.md": "cb9e1da38400b658f9209e465b4e5896fadd17dc66afa862f38bae8d8c1f2058" + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/check_capped_outer.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/check_capped_outer.py new file mode 100644 index 0000000000..83223e0a2b --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/check_capped_outer.py @@ -0,0 +1,131 @@ +#!/usr/bin/env python3 +"""Independent rational ledger for capped phase interiors; proof review separate.""" +from fractions import Fraction as Q +from decimal import Decimal, localcontext +from pathlib import Path +import json + +ROOT=Path(__file__).resolve().parent + +def decimal(x): + with localcontext() as ctx: + ctx.prec=65 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def calculate(q=None,zeta=None,sigma_boundary=None,old_vd=False,full_cell_gauge=False): + bit=Q('0.001733427039589844234749') + complex_=Q('0.001675462676808103522925') + q=complex_-Q(1,10**10) if q is None else q + a=(q+complex_)/2 + r=Q(1,10**6) + zeta=Q(1,10**8) if zeta is None else zeta + sigma_boundary=Q(1,10**5) if sigma_boundary is None else sigma_boundary + c=Q(3,32) + stop=Q(1,10**9) + eps=(1+q+q*(c-sigma_boundary-r-zeta))/(1+q)**2 + b=eps*(1+q)-c+sigma_boundary + beta=1-b-r-zeta + kappa=Q('0.001672922349588871') + selected=Q('0.001672660197076918') + tau,s=1-a,1-complex_ + internal=tau+(1-stop)*max(s-tau,Q(0)) + leaf=s+stop*(1-s) + lam=1-(q+a)/2 + key=beta+1-eps + margins=dict( + main_transform=eps*q, + global_coordinate_router=a, + fused_gaussian=1-eps+q*beta, + group_bit_router=1-eps+a*beta, + group_scans=1-eps+beta, + boundary_batched_algebra=c+b-eps, + sparse_union_movement=1-eps-beta-Q(1,2)+b, + sparse_union_full_key=1-eps-beta-Q(1,2)+b-key, + polynomial_products=eps, + microbox_face_corrections=1-(eps+beta+3-8+2), + ) + if old_vd: margins['unreplaced_per_axis_scan']=1-eps + slacks=dict( + actual_bit_above_effective=bit-a, + complex_above_effective=complex_-a, + complex_below_one_over_32=Q(1,32)-complex_, + q_positive=q, + complex_endpoint=complex_-q, + q_below_effective_bit=a-q, + stop_positive=stop, + stop_below_one=1-stop, + leaf_saving_above_effective_bit=(1-stop)*complex_-a, + q_below_internal=1-internal-q, + q_below_leaf=1-leaf-q, + lambda_above_tau=lam-tau, + lambda_above_sigma=lam-s, + lambda_above_internal=lam-internal, + lambda_prime_above_lambda=1-q-lam, + lambda_prime_above_leaf=1-q-leaf, + epsilon_positive=eps, + epsilon_below_one=1-eps, + capped_block_above_band=b-Q(1,2), + capped_block_below_cell=eps-b, + group_positive=beta, + group_below_dimension=eps-beta, + support_router_span=1-beta, + group_gauge_power_slack=1-b-r-beta, + actual_logarithmic_group_guard=1-b-beta, + global_gamma_power_slack=1-eps-r, + actual_logarithmic_gamma_guard=1-eps, + alpha_logarithm_absorption=r, + group_key_fits_record=1-key, + membership_arithmetic_fits_record=1-beta-2*(1-eps), + microbox_halo_volume=8-3-beta, + boundary_positive_slack=sigma_boundary, + short_record_fallback=eps-a, + improvement=kappa-selected, + ) + if full_cell_gauge: + slacks['unrecentered_group_gauge']=1-eps-r-beta + slacks.update({name+'_above_kappa':value-kappa for name,value in margins.items()}) + assert margins['main_transform']==margins['fused_gaussian'] + assert margins['boundary_batched_algebra']==margins['main_transform']+sigma_boundary + return dict(parameters=dict(a_bit=bit,a_eff=a,complex_saving=complex_,q=q, + r_spare=r,zeta=zeta,sigma_boundary=sigma_boundary,stop=stop, + epsilon=eps,capped_block_power=b,beta_group=beta, + omega_upper=Q(45,16),c_boundary=c,kappa=kappa,selected=selected, + lambda_=lam,lambda_prime=1-q,tau=tau,sigma=s, + microbox_power=Q(8),halo_power=Q(3)),margins=margins,slacks=slacks) + +def encode(o): + if isinstance(o,Q):return dict(rational=str(o),decimal=decimal(o)) + if isinstance(o,dict):return {k:encode(v) for k,v in o.items()} + return o + +if __name__=='__main__': + row=calculate() + assert 7**16<2**45 + assert all(value>0 for value in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + tests=[('endpoint_equality',dict(q=Q('0.001675462676808103522925'))), + ('zero_group_guard',dict(zeta=Q(0))), + ('zero_boundary_slack',dict(sigma_boundary=Q(0))), + ('old_per_axis_scan',dict(old_vd=True)), + ('unrecentered_full_cell_gauge',dict(full_cell_gauge=True))] + for name,kwargs in tests: + bad=calculate(**kwargs) + failed={k:v for k,v in bad['slacks'].items() if v<=0} + assert failed,name + controls[name]=failed + output=dict(status='PASS_CONDITIONAL_PHASE_FUSION_REVIEW', + scope='Local conditional transfer and exact arithmetic; inherited supplier and all-size interfaces remain assumptions. No upstream selected certificate is changed.', + review=dict(integrated_review='Root reviewed the complete standalone note, integrated proof chain, exact ledger, and independent cap/supplier/layout audits.', + independent_cap_gauge='CAPPED-GAUGE-AUDIT.md', + independent_cap_layout='CAPPED-SIGNOFF.md', + supplier_support='SUPPLIER-SUPPORT-AUDIT.md'), + **row,controls=controls) + (ROOT/'CAPPED-OUTER-ARITHMETIC.json').write_text(json.dumps(encode(output),indent=2)+'\n') + print(output['status']) + for name in ['epsilon','capped_block_power','beta_group','kappa']: + print(name,decimal(row['parameters'][name])) + print('minimum margin',decimal(min(row['margins'].values()))) + print('gain over selected',decimal(row['slacks']['improvement'])) + print('positive strict slacks',len(row['slacks'])) + print('controls',', '.join(controls)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/check_matrix_corollary.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/check_matrix_corollary.py new file mode 100644 index 0000000000..3f543803d7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/check_matrix_corollary.py @@ -0,0 +1,143 @@ +#!/usr/bin/env python3 +"""Exact corollary ledger, additionally assuming the supplied 9/4 theorem.""" +from fractions import Fraction as Q +from decimal import Decimal,localcontext +from pathlib import Path +import json + +ROOT=Path(__file__).resolve().parent + +def decimal(x): + with localcontext() as ctx: + ctx.prec=65 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def calculate(q=None,zeta=None,union_slack=None,old_vd=False, + full_cell_gauge=False,add_full_key_sort=False): + bit=Q('0.001733427039589844234749') + complex_=Q('0.001675462676808103522925') + q=complex_-Q(1,10**10) if q is None else q + a=(q+complex_)/2 + r=Q(1,10**6) + zeta=Q(1,10**8) if zeta is None else zeta + union_slack=Q(1,10**5) if union_slack is None else union_slack + tau_matrix=Q(19,8) + c=(3-tau_matrix)/2 + stop=Q(1,10**9) + cap=(Q(3,2)+q-(1+q)*(r+zeta)+union_slack)/(2+q) + eps=(1+q*(1-cap-r-zeta))/(1+q) + beta=1-cap-r-zeta + kappa=Q('0.001673360105405778') + selected=Q('0.001672660197076918') + first=Q('0.001672922349588871') + tau_bit,sigma=1-a,1-complex_ + internal=tau_bit+(1-stop)*max(sigma-tau_bit,Q(0)) + leaf=sigma+stop*(1-sigma) + lam=1-(q+a)/2 + key=beta+1-eps + margins=dict( + main_transform=eps*q, + global_coordinate_router=a, + fused_gaussian=1-eps+q*beta, + group_bit_router=1-eps+a*beta, + group_scans=1-eps+beta, + boundary_algebraic_matrix=c+cap-eps, + sparse_union_movement=1-eps-beta-Q(1,2)+cap, + polynomial_products=eps, + microbox_face_corrections=1-(eps+beta+3-8+2), + ) + if old_vd:margins['unreplaced_per_axis_scan']=1-eps + if add_full_key_sort: + margins['unnecessary_union_full_key_sort']=margins['sparse_union_movement']-key + slacks=dict( + actual_bit_above_effective=bit-a, + complex_above_effective=complex_-a, + complex_below_one_over_32=Q(1,32)-complex_, + q_positive=q, + complex_endpoint=complex_-q, + q_below_effective_bit=a-q, + stop_positive=stop, + stop_below_one=1-stop, + leaf_saving_above_effective_bit=(1-stop)*complex_-a, + q_below_internal=1-internal-q, + q_below_leaf=1-leaf-q, + lambda_above_tau=lam-tau_bit, + lambda_above_sigma=lam-sigma, + lambda_above_internal=lam-internal, + lambda_prime_above_lambda=1-q-lam, + lambda_prime_above_leaf=1-q-leaf, + epsilon_positive=eps, + epsilon_below_one=1-eps, + capped_block_above_band=cap-Q(1,2), + capped_block_below_cell=eps-cap, + group_positive=beta, + group_below_dimension=eps-beta, + support_router_span=1-beta, + group_gauge_power_slack=1-cap-r-beta, + actual_logarithmic_group_guard=1-cap-beta, + global_gamma_power_slack=1-eps-r, + actual_logarithmic_gamma_guard=1-eps, + alpha_logarithm_absorption=r, + group_key_fits_record=1-key, + membership_arithmetic_fits_record=1-beta-2*(1-eps), + sparse_union_fraction_decays=cap-beta-Q(1,2), + microbox_halo_volume=8-3-beta, + union_positive_slack=union_slack, + matrix_exponent_above_source_bound=tau_matrix-Q(9,4), + matrix_exponent_below_five_halves=Q(5,2)-tau_matrix, + short_record_fallback=eps-a, + improvement_over_selected=kappa-selected, + improvement_over_first_reviewed_transfer=kappa-first, + ) + if full_cell_gauge: + slacks['unrecentered_group_gauge']=1-eps-r-beta + slacks.update({name+'_above_kappa':value-kappa for name,value in margins.items()}) + assert margins['main_transform']==margins['fused_gaussian'] + assert margins['sparse_union_movement']==margins['main_transform']+union_slack + return dict(parameters=dict(a_bit=bit,a_eff=a,complex_saving=complex_,q=q, + r_spare=r,zeta=zeta,sigma_union=union_slack,stop=stop, + epsilon=eps,capped_block_power=cap,beta_group=beta, + matrix_exponent=tau_matrix,matrix_source_bound=Q(9,4), + c_boundary=c,kappa=kappa,selected=selected,first_reviewed_transfer=first, + lambda_=lam,lambda_prime=1-q,tau=tau_bit,sigma=sigma, + microbox_power=Q(8),halo_power=Q(3)),margins=margins,slacks=slacks) + +def encode(o): + if isinstance(o,Q):return dict(rational=str(o),decimal=decimal(o)) + if isinstance(o,dict):return {k:encode(v) for k,v in o.items()} + return o + +if __name__=='__main__': + row=calculate() + assert all(value>0 for value in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + tests=[('endpoint_equality',dict(q=Q('0.001675462676808103522925'))), + ('zero_group_guard',dict(zeta=Q(0))), + ('zero_union_slack',dict(union_slack=Q(0))), + ('old_per_axis_scan',dict(old_vd=True)), + ('unrecentered_full_cell_gauge',dict(full_cell_gauge=True)), + ('full_key_sort_on_union',dict(add_full_key_sort=True))] + for name,kwargs in tests: + bad=calculate(**kwargs) + failed={k:v for k,v in bad['slacks'].items() if v<=0} + assert failed,name + controls[name]=failed + output=dict(status='PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW', + scope='Additional matrix-conditioned corollary. Assumes supplied omega_C<=9/4 theorem and the effective algebraic stability transfer, as well as all first-transfer assumptions.', + review=dict(integrated='Root read the matrix corollary and independent layout signoff, reproduced the exact parameters and ledger, and replayed matrix source construction controls.', + matrix_source='PROOF-AND-COVERAGE.md', + effective_stability='ALGEBRAIC-STABILITY-INDEPENDENT.md', + independent_layout='MATRIX-LAYOUT-SIGNOFF.md'), + omitted_row=dict(name='optional full-coordinate-key union surcharge', + reason='The actual U_i complete-rectangle extraction, one-field transpose, and ordered merge do not sort union records. Sparse microhalo keys remain paid in their separate cost.'), + **row,controls=controls) + (ROOT/'MATRIX-COROLLARY-ARITHMETIC.json').write_text(json.dumps(encode(output),indent=2)+'\n') + print(output['status']) + for name in ['epsilon','capped_block_power','beta_group','kappa']: + print(name,decimal(row['parameters'][name])) + print('minimum margin',decimal(min(row['margins'].values()))) + print('gain over selected',decimal(row['slacks']['improvement_over_selected'])) + print('gain over first transfer',decimal(row['slacks']['improvement_over_first_reviewed_transfer'])) + print('positive strict slacks',len(row['slacks'])) + print('controls',', '.join(controls)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/check_outer.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/check_outer.py new file mode 100644 index 0000000000..2bd3f6eb68 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/check_outer.py @@ -0,0 +1,131 @@ +#!/usr/bin/env python3 +"""Exact arithmetic only. This does not automatically certify proof obligations.""" +from fractions import Fraction as Q +from decimal import Decimal, localcontext +from pathlib import Path +import json + +ROOT = Path(__file__).resolve().parent + +def dec(x): + with localcontext() as ctx: + ctx.prec = 65 + return str(Decimal(x.numerator) / Decimal(x.denominator)) + +def compute(q=None, zeta=None, old_vd=False): + b = Q('0.001675462676808103522925') + a_bit = Q('0.001733427039589844234749') + q = b-Q(1,10**10) if q is None else q + r = Q(1,10**6) + zeta = Q(1,10**8) if zeta is None else zeta + a_eff = (q+b)/2 + t_stop = Q(1,10**9) + tau, sigma = 1-a_eff, 1-b + lam_prime = 1-q + lam = 1-(q+a_eff)/2 + internal = tau + (1-t_stop)*max(sigma-tau,Q(0)) + leaf = sigma+t_stop*(1-sigma) + eps = (1+q-q*(r+zeta))/(1+2*q) + beta = 1-eps-r-zeta + kappa = Q('0.001672664764353193') + selected = Q('0.001672660197076918') + omega_upper = Q(45,16) + # alpha^2=O(log p), so w=O(sqrt(p) polylog(p)). + wide = Q(1,2) + key = beta+1-eps + margins = dict( + main_transform=eps*q, + global_coordinate_router=a_eff, + fused_gaussian=1-eps+q*beta, + group_bit_router=1-eps+a_eff*beta, + group_scans=1-eps+beta, + boundary_batched_algebra=1-(omega_upper-1)*wide, + sparse_union_movement=1-beta-wide, + sparse_union_full_key=1-beta-wide-key, + polynomial_products=eps, + # gA/L copies, g sequential passes, each p^2 conservatively. + microbox_face_corrections=1-(eps+beta+3-8+2), + ) + if old_vd: + margins['unreplaced_per_axis_scan'] = 1-eps + slacks = dict( + actual_bit_above_effective=a_bit-a_eff, + complex_above_effective=b-a_eff, + complex_below_one_over_32=Q(1,32)-b, + q_positive=q, + complex_endpoint=b-q, + q_below_effective_bit=a_eff-q, + stop_positive=t_stop, + stop_below_one=1-t_stop, + leaf_saving_above_effective_bit=(1-t_stop)*b-a_eff, + q_below_internal=1-internal-q, + q_below_leaf=1-leaf-q, + lambda_above_tau=lam-tau, + lambda_above_sigma=lam-sigma, + lambda_above_internal=lam-internal, + lambda_prime_above_lambda=lam_prime-lam, + lambda_prime_above_leaf=lam_prime-leaf, + lambda_prime_below_one=q, + epsilon_positive=eps, + epsilon_below_one=1-eps, + group_positive=beta, + group_below_dimension=eps-beta, + group_controls_below_one_axis_width=1-eps-beta, + support_router_span=1-beta, + group_guard=1-eps-r-beta, + alpha_logarithm_absorption=r, + local_gauge_sublinear=1-eps-r, + core_above_band=eps-wide, + record_suffix=1-eps, + prime_interval_packing=1-eps, + semantic_dimension_guard=1-eps, + microbox_larger_than_halo=Q(8-3)-beta, + short_record_fallback=eps-a_eff, + improvement=kappa-selected, + ) + slacks.update({name+'_above_kappa':val-kappa for name,val in margins.items()}) + assert eps*q == 1-eps+q*beta + return dict(parameters=dict(a_bit=a_bit,a_eff=a_eff,b=b,q=q,r=r,zeta=zeta, + t_stop=t_stop,tau=tau,sigma=sigma,lambda_=lam,lambda_prime=lam_prime, + epsilon=eps,beta_group=beta,kappa=kappa,selected=selected, + omega_upper=omega_upper,microbox_power=Q(8),halo_power=Q(3)), + margins=margins,slacks=slacks) + +def convert(o): + if isinstance(o,Q): return {'rational':str(o),'decimal':dec(o)} + if isinstance(o,dict): return {k:convert(v) for k,v in o.items()} + return o + +if __name__ == '__main__': + row=compute() + assert 7**16 < 2**45 + assert all(x>0 for x in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + for name,kw in [('complex_endpoint_equality',{'q':Q('0.001675462676808103522925')}), + ('zero_group_guard',{'zeta':Q(0)}), + ('old_per_axis_scan',{'old_vd':True})]: + bad=compute(**kw) + failed={n:v for n,v in bad['slacks'].items() if v<=0} + assert failed, name + controls[name]={'rejected_slacks':failed} + result=dict(status='PASS_EXACT_ARITHMETIC_PENDING_FULL_PROOF_REVIEW', + scope='Exact outer arithmetic; no automatic theorem promotion.', + **row, controls=controls, + theorem_obligations=[ + 'retained finite suppliers and exact common-frame/mutable-source compiler', + 'deferred reservations and paid global synthetic layout', + 'shallow Toeplitz plus SPD extension and tensor compiler', + 'small-a forward conditioning and exact target-grid square completion', + 'grouped sparse-union route and batched boundary arithmetic', + 'dyadic core with sparse microbox-face repair and monotone q-image scans', + 'scalar product-diagonal scan and retained all-size semantic/prime/error transfer', + ]) + path=ROOT/'OUTER-ARITHMETIC.json' + path.write_text(json.dumps(convert(result),indent=2)+'\n') + print(result['status']) + print('kappa',dec(row['parameters']['kappa'])) + print('minimum margin',dec(min(row['margins'].values()))) + print('gain',dec(row['slacks']['improvement'])) + print('positive exact slacks',len(row['slacks'])) + print('negative controls',', '.join(controls)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/AUDIT.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/AUDIT.md new file mode 100644 index 0000000000..bbeeee7431 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/AUDIT.md @@ -0,0 +1,242 @@ +# Grouped phase-cell layout audit + +The sparse-union layout in `REFINED-PHASE-FUSION.md` has a constructive +implementation with its stated `V g^2 W polylog(p)/d` per-group movement +budget. It does not require sorting the full tensor or retaining d full +payload scans. The conclusion is conditional on the separately supplied +dense local tensor compiler, rounded phase-cell factorization, and precision +bounds. This audit does not prove those algebraic/analytic inputs. + +The maximum physical bandwidth **W**, including the broad forward map, must +be used in the dyadic carving and support sets below. The narrower inverse +bandwidth alone is insufficient for the broad map. The phase-assembly +branch confirms this convention. + +## Exact operator order and the union that is actually invariant + +Write H in interior/boundary order as `[A B; C D]` and let +`S=D-C A^-1 B`. The displayed identity in the parent note is correct. In +execution order its inverse can be scheduled as + +``` +diag(A^-1,I) +[I 0; -C I] +diag(I,S^-1) +diag(A,I) +[I -B; 0 I] +diag(A^-1,I). +``` + +The middle two diagonal factors commute. Thus there are three dense +interior stages and two sparse windows: the C and S^-1 stages share one +window; B uses the second window. For H itself, reverse and invert this +sequence: the same original B,C and S occur, with signs reversed and dense +A/A^-1 interchanged. No dense A^-1 B or C A^-1 shear is smuggled into the +sparse boundary cost. + +For each active axis i, let J_i be its designated boundary coordinates. +Let E_i contain J_i and every interior coordinate incident to an original +nonzero entry of B_i or C_i. Original physical bandwidth W implies that E_i +lies within O(W) of the carved endpoints. Include physical-period cuts, +artificial phase cuts and clipped microbox endpoints. Dyadic carving gives + +``` +|E_i| <= C (L/d+1) W log d, +mu := max_i |E_i|/L = O(W log d/d + W log d/L) +``` + +inside a length-L microbox. All sparse factors have the block form +`T_i = I_(E_i^c) direct_sum T_i|E_i`. In particular S_i^-1 can be dense on +J_i: J_i is already included in E_i. Let + +``` +U = {x in [L]^g : some x_i belongs to E_i}. +``` + +Then `|U|/L^g <= g mu`. Every sparse tensor-axis operation preserves U and +its complement. If x_i is changed by a nonzero off-diagonal entry, both +its old and new coordinates belong to E_i; otherwise it is unchanged. +This proves closure for an arbitrarily long sequence of these sparse +operations, not just one application. + +Dense A or A^-1 is **not** covered. It can move a near-boundary interior +coordinate into the deep interior. Scatter U before a dense stage and +regather afterwards. `check_union.py` includes an exact example in which +dense A^-1 sends union-supported data outside U. The actual schedule above +therefore uses two union gathers, not one retained union for all six stages. + +## Complete rectangles inside the union: no missing-zero assumption + +For one sparse axis i define + +``` +U_i = {x : x_i in E_i} = E_i x [L]^(g-1). +``` + +This is a **complete rectangle**, and it is wholly present in U regardless +of which other coordinates are near a boundary. All nonidentity data needed +by T_i is in U_i. A line with every other coordinate outside its own E_j +still has every E_i coordinate present. A line with another boundary +coordinate may contain more data, but those extra E_i^c entries are identity +coordinates and need no boundary computation. + +Keep a copy of U in its original microbox lexicographic order, with a short +coordinate/membership header. For each axis i: + +1. Scan U once and copy records with x_i in E_i to an auxiliary stream. + The copied stream is lexicographic E_i x [L]^(g-1). +2. Rename the increasing E_i coordinates by their ordinal 0,...,|E_i|-1. + This is a stream counter; no rank/select query is needed. +3. Expose that one field at the line suffix using the retained small-field + transpose. Its length is at most L=poly(p), so only O(log p) binary + block moves are needed. Pad only this field to its next power of two + temporarily, at a factor below two, and remove that padding before the + next axis. The other fields may be arbitrary spectators. +4. Apply the original B/C or boundary S/S^-1 factor on its complete E_i + line, retaining identity coordinates E_i\J_i when appropriate. +5. Undo the transpose at the actual field length. Merge the ordered E_i + stream back into a fresh copy of U, replacing exactly those entries. + +No array is completed by invented zero coordinates. The finite checker +verifies the extracted rectangle cardinality and the full tensor result. + +The extraction/merge costs O(|U|) per axis. Transposition costs +O(|U_i| polylog p). Consequently one sparse window costs + +``` +O(V [g^2 mu + g mu polylog p]) + = O(V g^2 W polylog p/d) + O(V g^2 W polylog p/L). +``` + +There is no factor `K=g log s` from a computed-key sort in this part of the +algorithm. A more general radix implementation would still fit with extra +small powers of g, but the complete-rectangle construction avoids it. + +## Union extraction and piece catalogues + +Only one full microbox scan is needed to form U, and one to scatter it. +For every complete p-bit coefficient record, compute its g membership flags +from its coordinate tuple and OR them. The flags and coordinate header have +O(g log L) bits. A nonzero number of flags sends the whole record once to U; +it is not duplicated for each active axis. + +This is a charged descriptor calculation. For actual source periods of +bitlength h=Theta(p/d), q_j inversion and phase-cell endpoint calculations +use O(h^2 polylog p) elementary bit operations per axis. Within a phase +cell, the greedy dyadic carve is found by at most O(log d) halvings and +integer comparisons. Thus a conservative membership cost is +`O(g h^2 polylog p)` per record, which is o(p) for +`g=p^(1-epsilon-r-zeta)` and epsilon>2/3. At the proposed epsilon this has +power about 0.005. It is absorbed by reading the p-bit record once. +Alternatively the original remainder phase generator and an odometer give +the same membership flags in sequential coordinate order. + +Factor pages may be regenerated once per complete microbox/group from its +actual coarse origins, using the supplied polynomial-time coefficient +generation. The complete inner microbox and untouched spectator payload +are superpolynomial in p and amortize this polynomial setup. This avoids +random access to a global catalogue in the scrambled coarse order created +by a support-limited gather. Online row/factor pages are then visited in +the ordered extracted E_i line. All page rewinds are within that line or +RHS batch and are included in the corresponding boundary arithmetic. + +## Batched right-hand sides do not add a large transpose + +All other inner coordinates and following tensor axes are independent +right-hand sides for an axis operation. Their count R is superpolynomial. +Batch them in a dyadic width between W and 2W; the final incomplete batch +can be padded once at constant cost. To put the batch index outside the +boundary-row field, move only that polynomial-width row field. This costs +O(log p) times the boundary volume, not a transpose of the long spectator +field. Each W-by-W factor page is scanned once per RHS batch and reused in +the block product. Reverse the small-field layout after the solve. + +Thus the assumed Strassen `W^omega` per W RHS gives the quoted total +`O(V W^(omega-1) polylog p)` boundary arithmetic. This audit checks the +layout and factor reuse; stability and the no-pivot factor theorem remain +the phase-assembly branch's responsibility. + +## q_j embeddings, selections and source order need one group scan + +For a group, each coordinate map `j -> q_j=floor(tj/s+1/2)` is increasing +and injective. Its tensor product preserves lexicographic order on its +image. Enumerate the target-group rectangle once. At target coordinate k, +set + +``` +j=max(0,ceil(s(2k-1)/(2t))). +``` + +The coordinate belongs to the image iff `j infinity`. Individual guarded neighborhoods +fit the original long axis; the collection of all g controls does not have +to fit in a single axis. This is the same support proof already audited. + +Microedge correction is unchanged. The underlying rounded operator and its +physical bandwidth do not change when a factorization receives more cuts. +Only the principal restriction and Schur decomposition are repartitioned. +The retained Neumann/principal-window comparison still applies to that +same operator. With L=p^8 and halo A=p^3, the additional output fraction is +O(g p^-5), now about p^-4.9064. Its sparse gather, true-period correction and +old sequential local solvers are still negligible. Capping does not turn +this into a dense non-dyadic tensor halo. + +The pinned independent arithmetic checks give approximately + +``` +epsilon = .9984838711085558 +b = .9064167934681446 +beta = .0935821965318554 + +per-group union movement power = -.2192524004044339 +total union movement power = .6856492741722665 +total group scan power = .9049016745767004 +membership descriptor power = .0966144543147439. +``` + +`check_capped_layout.py` independently checks finite capped partitions, +dyadic block sizes, separator distances, worst-pattern elimination fill, +and these exact rational layout inequalities against the pinned candidate +JSON. No layout blocker was found. + +This signoff is specifically for carving, sparse support, actual stream +movement, descriptors, and inherited microedge placement. It does not +independently prove the newly anchored broad/narrow Gaussian formulas or +their final precision bound. Those remain the algebraic/analytic cap audit. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MANIFEST.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MANIFEST.json new file mode 100644 index 0000000000..1fd05118b7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MANIFEST.json @@ -0,0 +1,16 @@ +{ + "AUDIT.md": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", + "CAPPED-LAYOUT-CONTROLS.json": "6c1342c436fb487a5aa39cf563d2a2e2fddd94d3e441a6f4889cb1e2b73a478c", + "CAPPED-SIGNOFF.md": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a", + "MATRIX-LAYOUT-CONTROLS.json": "f8bd8729cb8438188e90fb9149f598b825461c97b4ed290584ca4367f812a48a", + "MATRIX-LAYOUT-SIGNOFF.md": "4c8c62f723232e842dc3bbdcc4defcc8ff1aeb86ba9f694661899dc6783b13c4", + "PARAMETERS.json": "9f72cfffa34844764ca65a6b4431cc5c779141751df746019db9d3ad8f1dee92", + "Q-EMBEDDING-CONTROLS.json": "24ead3534482397c00c23ad3dbbd4e930e02290e00c97765353d8783c6575c7d", + "README.md": "0ec721f76dfa943e808afd3c2cc39ee69f1a6e5f4f77ba3c7c1373ad60db25af", + "UNION-CONTROLS.json": "6c5262144215cc603bbc3bee93a9a6d9d2318d0177a59115fe560762998498f9", + "check_capped_layout.py": "267d93ab5114ec160f88a6e65fd34c9f0696a686c36ba2dd194d6b70fa32a6bd", + "check_q_embedding.py": "ac858a14bbaf37e0e412a7ada282cd1586632f52b8a979733e775f50130969b6", + "check_union.py": "74608380d72567edcd4470c8d226b15848f9d3a4e5d77d652bdd8963480ab9ce", + "q-embedding.log": "24ead3534482397c00c23ad3dbbd4e930e02290e00c97765353d8783c6575c7d", + "union.log": "6c5262144215cc603bbc3bee93a9a6d9d2318d0177a59115fe560762998498f9" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json new file mode 100644 index 0000000000..aee3270196 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json @@ -0,0 +1,30 @@ +{ + "status": "PASS_MATRIX_COROLLARY_NO_SORT_LAYOUT", + "source_sha256": "b0d4648e16df74fc0a3644baa8a82cbb82842c0aa21abe915a28c0c4c949182a", + "exponents": { + "union_fraction": { + "rational": "-3430918384747396741111767383/8006701850307232414091700000000", + "decimal": "-0.000428505825356260251146382720850643577327042013532977151967005" + }, + "membership": { + "rational": "269791328968194081006854064986174254715733926/1069348903949477650676785089185287127357866963", + "decimal": "0.252294950667420906279768139101314143625086172719573255573065" + }, + "total_union": { + "rational": "320264641396534864583914893734048954038319544983689789/320804671184843295203035526755586138207360088900000000", + "decimal": "0.998316639884594221546182952048279623715798111123297150773509" + }, + "group_scans": { + "rational": "2402698356122892965636680000000000000000000000/3208046711848432952030355267555861382073600889", + "decimal": "0.748959903622628611922902526129555589081788674143596616501459" + }, + "microhalo_fraction": { + "rational": "-380335532915361620704585/80067018503072324140917", + "decimal": "-4.75021475791267813012557319136042532178866352100676648857598" + }, + "rejected_union_full_key": { + "rational": "3334408963974161672450196077539/2668900616769077471363900000000", + "decimal": "1.24935673626196560962328042591872403463400943697970053427205" + } + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md new file mode 100644 index 0000000000..d683e4d495 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md @@ -0,0 +1,64 @@ +# Matrix-corollary layout: positive signoff + +The no-sort union layout in `AUDIT.md` applies unchanged to +`work/phase_assembly_ultra/MATRIX-COROLLARY.md`. Its algebraic conditions are +closed axis supports E_i and a complete microbox; neither its proof nor its +implementation requires g2*W and B&(B-1)==0 and n>=2*W + J=set(range(W))|set(range(n-W,n));blocks=[];c=W;stop=n-W + while stop-c>=B+W: + blocks.append(list(range(c,c+B)));J.update(range(c+B,c+B+W));c+=B+W + while stop-c>4*W: + rem=stop-c;k=1<<((rem-2*W).bit_length()-1) + assert k<=B + J.update(range(c,c+W));blocks.append(list(range(c+W,c+W+k))) + J.update(range(c+W+k,c+2*W+k));c+=2*W+k + J.update(range(c,stop)) + assert sorted(J|{j for b in blocks for j in b})==list(range(n)) + assert sum(map(len,blocks))+len(J)==n + assert all(len(b)<=B and len(b)&(len(b)-1)==0 for b in blocks) + assert all(b[0]-a[-1]>W for a,b in zip(blocks,blocks[1:])) + assert len(J)<=10*W*((n+B-1)//B+B.bit_length()+1) + return blocks,J + +def fill_bound(n,blocks,J,W): + adj=[set(j for j in range(n) if j!=i and abs(i-j)<=W) for i in range(n)] + for block in blocks: + for v in block: + nbr=list(adj[v]) + for x in nbr: + adj[x].update(y for y in nbr if y!=x);adj[x].discard(v) + adj[v].clear() + order={v:i for i,v in enumerate(sorted(J))} + bandwidth=max((abs(order[i]-order[j]) for i in J for j in adj[i]),default=0) + assert bandwidth<=2*W + return bandwidth + +def main(): + total=0;examples=[] + for W in (1,2,3,4): + for B in (8,16,32,64): + if B<=2*W:continue + for n in range(2*W,4*B+1): + blocks,J=carve(n,B,W);total+=1 + n=4*B+7;blocks,J=carve(n,B,W);bw=fill_bound(n,blocks,J,W) + examples.append(dict(W=W,B=B,n=n,blocks=len(blocks),boundary=len(J),schur_boundary_bandwidth=bw)) + raw=SOURCE.read_bytes();source=json.loads(raw);p={k:Q(v['rational']) for k,v in source['parameters'].items()} + e,b,g=p['epsilon'],p['capped_block_power'],p['beta_group'];w=Q(1,2);h=1-e + exponents=dict(per_group_union_movement=2*g+w-b,total_union_movement=e+g+w-b, + total_group_scans=e-g,metadata_membership=g+2*h,microbox_key=g, + global_sparse_key=g+h,guarded_support_span=g, + sparse_microhalo_volume=g+3-8, + hypothetical_fullkey_union=e+2*g+h+w-b) + assert Q(1,2)1, theta=rho-1, q_i=floor(rho i+1/2), beta_i=rho i-q_i for all integer i. Define + +``` +x_i=exp(2a rho q_i), e_i=exp(-a rho^2 i^2+a beta_i^2), +G_ij=exp(-a q_i^2) x_i^j e_j, +r=exp(2a rho), c=product_(m>=1)(1-r^-m)>0. +``` + +The sequence x_i increases at least by a factor r at every step and tends geometrically to zero toward minus infinity. + +## Finite-origin identities + +For an integer origin l, Newton interpolation of columns j>=l gives, on every finite principal interval starting at l, + +``` +C_i(l)=product_(l<=k=j, +U_ij(l)=(e_j/e_i) h_(j-i)(x_l,...,x_i), j>=i, +G=L(l) diag(C_i(l)) U(l). +``` + +Here h denotes complete homogeneous symmetric polynomials, and both triangular factors have diagonal one. The powers arising from moving the monomial origin from zero to l cancel exactly: the pivot is C_i(l), not a residual power of x_i. + +Since C_i(l)>=c and the normalized complete-homogeneous coefficient is at most c^-1, + +``` +|L_ij(l)| <= c^-1 G_ij, +|U_ij(l)| <= c^-1 G_ij. +``` + +All limits as l tends to minus infinity exist, including fixed-degree h over the infinite summable past. The displayed Gaussian domination permits passing the factorization to the bilateral limit. + +## Inverse factors + +For i>=j and h=i-j, + +``` +(L(l)^-1)_ij=(-1)^h C_i(l)/C_j(l) + * exp(a[-beta_i^2+beta_j^2-rho^2 h+2rho sum_(j=i and h=j-i, + +``` +(U(l)^-1)_ij=(-1)^h (e_j/e_i) e_h(x_l,...,x_(j-1)). +``` + +The latter e_h is elementary symmetric, distinct from the scalar normalization e_j. Its dominant product consists of x_i,...,x_(j-1), so its normalized value is at most c^-1. + +Using |beta_i|<=1/2 in the two exact exponents gives uniform estimates + +``` +|(L(l)^-1)_ij| <= c^-2 exp(a/4-a rho theta h), +|(U(l)^-1)_ij| <= c^-1 exp(a/4-a rho theta h). +``` + +These pass to the bilateral limits. Products of triangular factors and their inverse factors have only finitely many terms for each supported matrix entry, so both inverse identities survive directly. All four factors define bounded absolutely row-summable operators. + +## Periodization + +Every limiting factor is simultaneously s-periodic in its row and column indices. For a bounded absolutely row-summable periodic array A, define + +``` +Pi(A)_ij=sum_(n in Z) A_(i,j+ns), 0<=i,j=1**, this is O(theta^-2), with a constant depending only on a. Thus a fixed admissible integer Gaussian width alpha suffices for the norm and perturbation estimates. No logarithmic growth of alpha is needed for conditioning. + +The theorem changes an analytic parameter constraint. Applying these factors quickly on the tape remains a separate constructive obligation. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md new file mode 100644 index 0000000000..59d9128d75 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md @@ -0,0 +1,211 @@ +# Independent checks of Gaussian interpolation and holes completion + +Status: exact algebraic identities and analytic bounds below; no complete new inverse implementation, and no final kappa claim. This continues the parent investigation after the first outer-reduction package. + +## 1. Local scaled Vandermonde identity + +Write `a=pi*alpha^2`, `rho=t/s`, `q_i=floor(rho*i+1/2)`, and `beta_j=rho*j-q_j`. On any nonwrapping index interval, + +``` +N_ij = exp[-a*(rho*j-q_i)^2 + a*beta_j^2] + = exp[-a*q_i^2] * (exp[2a*rho*q_i])^j + * exp[-a*rho^2*j^2+a*beta_j^2]. +``` + +Translations of both index origins preserve a diagonally scaled Vandermonde representation. If the local span is L, centered nodes have logarithms O(aL), and the product-tree polynomials, Lagrange denominators, and outer diagonals have logarithmic height O(aL^2+L). Thus L=O(sqrt(p/a)) is compatible with Cp-bit working precision. Distinct nodes are explicitly separated, so the inverse of a *single supplied principal block* can be computed by multipoint interpolation with that precision and O(L p polylog p) bit cost. + +This does not justify independent short blocks. The current inverse has strong directed coupling near the end of a phase cell. Its exact q-LDU below shows why a halo of width merely sqrt(p/a) can lose much more than 2^-p. + +## 2. Exact Gaussian Toeplitz LDU in a no-gap phase cell + +Within a cell `q_i=i+c`, put `theta=rho-1`, + +``` +q=exp(-a*rho), Q=q^2, +P_j=product_(k=1..j) (1-Q^k), P_0=1, +D_i=exp(a*theta*i^2-2a*c*i), +G_ij=q^((i-j)^2). +``` + +Then `N=diag(D_i)*G*diag(D_i)^-1`. With the Gaussian binomial + +``` +{i choose j}_Q = P_i/(P_j*P_(i-j)), +``` + +the exact lower factor and its inverse are + +``` +L_ij = q^((i-j)^2) * {i choose j}_Q, i>=j, +Linv_ij = (-q)^(i-j) * {i choose j}_Q, i>=j, +G = L * diag(P_j) * L^T. +``` + +These follow from the ordinary Newton factorization of the Vandermonde matrix at nodes Q^-i; eliminating its successive monic Newton polynomials gives the displayed Gaussian binomial factors and pivot P_j. The inverse follows from the finite q-binomial theorem, or multiplying the two triangular factors and grouping equal powers of Q. `check_q_ldu.py` verifies both identities exactly at 30 rational q/size choices (1,155 matrix entries), including diagonal conjugations. + +Each triangular application is a diagonally scaled Toeplitz convolution: + +``` +L = diag(P_i) T_(q^(h^2)/P_h) diag(P_i)^-1, +Linv = diag(P_i) T_((-q)^h/P_h) diag(P_i)^-1. +``` + +The N factors add only the D_i conjugations. Their inverse lower entries have exponent + +``` +a*((2*beta_i-rho)*h - theta*h^2), h=i-j. +``` + +Since beta_i<=1/2, + +``` +abs((D Linv D^-1)_ij) + <= C exp[-a*theta*h*(h+1)], +C = product_(k>=1)(1-Q^k)^-1. +``` + +The upper inverse obeys the same bound using beta_i>=-1/2. For a>=pi and rho>1, C<2 by a geometric bound on the logarithm of the infinite product. Hence both inverse triangular factors have row norm O(1+(a theta)^-1/2). The P_j diagonal is bounded between 1/C and 1. This avoids using cancellation of exponentially large factors to bound the *completed* inverse. Over one cell, the explicit D_i range costs O(a/theta) guard bits, so the retained condition alpha^2/theta=O(p) still suffices for a straightforward implementation. + +This supplies a clean alternative to the retained Gohberg–Semencul interior solve. The latter already has linear convolution cost; a new exponent requires treating the gaps between cells or their boundary system. + +## 3. Exact complementary hole positions + +Let 11, x_k=r^k, and V_t=(x_k^j)_(0<=k,j=k, and 0 otherwise, +(V_holes^T)_(j,h) = x_h^j. +``` + +The coefficient of X^(t-1-j) in P(X)/(X-x_h) equals +`sum_(k=0..j) p_(t-k) x_h^(j-k)`, by synthetic division from the leading end. Lagrange interpolation gives the identity. + +Given values only on selected nodes, set the H missing values initially to zero and do one full interpolation. Solve A for the missing values that cancel the H degree-tail coefficients, then interpolate with the completed values. The resulting polynomial has degree=1, + +``` +abs(B_ij) <= exp[-a*rho*(rho'*k^2-k)]. +``` + +Indeed the relevant distance is at least rho'k-1/2, and the subtractive delta_j^2 is at most 1/4. Thus off-diagonal row sums are bounded by a convergent geometric Gaussian tail, with first exponent a rho(rho'-1). For small theta, rho' is Theta(1/theta), so this exponent is Theta(a/theta). To obtain error 2^-p a direct truncated solve only needs complementary width + +``` +O(1+sqrt(p/(a*rho*rho'))) = O(1+sqrt(p theta/a)). +``` + +This rigorously confirms the conditioning and narrow-band part of the parent's holes-completion proposal. It is not a cost proof for the complete original inverse: the Toeplitz T and full geometric interpolation must first be implemented with their enormous diagonal scalings canceled symbolically. + +## 6. A useful normalized erasure-coefficient bound + +Let hole exponents h_0<...=g, let R=r^g>1, and e_k be the kth elementary symmetric polynomial in r^h. Let M_k be the product of the k largest nodes. Then + +``` +1 <= e_k/M_k <= product_(j>=1)(1-R^-j)^-1. +``` + +For any selected k-index subset, compare its indices with the k largest indices. Its nonnegative deficits form a partition, and its product divided by M_k is at most R to the negative partition size. Summing over partitions proves the bound. The contribution from deficit >=m is at most + +``` +R^(-m/2) * product_(j>=1)(1-R^(-j/2))^-1. +``` + +For R>=4 the latter product is below 4, so retaining deficit less than `m >= (2p+4)/log2 R` suffices for relative error below 2^-p. Thus normalized coefficients have bounded height even when unnormalized coefficients contain Theta(a theta t^2) exponent bits. Only O(p theta/a) partition energy near the subset boundary is relevant. + +Likewise, for a nonhole i, pull `product_h max(r^i,r^h)` out of |E(r^i)|. The residual is `product_h(1-r^-|i-h|)`, which lies between `product_(k>=1)(1-r^-k)^2` and 1. It is uniformly bounded away from zero for fixed r>1. These are positive products and sums; the bounds do not depend on fortuitous numerical cancellation. + +The unresolved algorithmic question is performing the needed polynomial multiplication/division while storing these *degree-dependent* scales, rather than one common scale. Feeding the raw coefficients to an ordinary FFT requires Theta(a theta t^2) bits; declaring that cost absent is invalid. The positive partition estimates give a concrete starting point for a banded or locally normalized implementation. + +## 7. Correct boundary of the result + +- Exact q-LDU, exact hole indexing, exact completion, complementary conditioning, and the normalized erasure bounds above are established algebraically. +- `Q-LDU-CONTROLS.json` and `HOLES-CONTROLS.json` contain finite independent checks. +- No stable all-size global holes solver is implemented here. Its fully paid precision/tape cost remains open. +- No final multiplication exponent follows from these identities alone. + +## 8. Weighted full-geometric interpolation: a stable local implementation + +This is a further positive lemma for one component of holes completion. It does **not** complete the Toeplitz/complement adapters. + +Let beta=H/t=theta/rho, let h_l be the actual holes above, and let h*_l=(l+1/2)/beta be the ideal equally spaced holes. For x in [0,t], define + +``` +R(x) = 2a*rho * sum_l (x-h_l)_+ - a*theta*x^2. +``` + +This is the logarithm of the dominant erasure evaluation after canceling the full-geometric Gaussian input diagonal, up to a constant independent of x. That constant can be kept symbolically and canceled at the final adapter. + +For ideal holes, put m=floor(beta*x+1/2). Then + +``` +sum_l (x-h*_l)_+ = m*x - m^2/(2beta), +R*(x) = -(a*rho/beta)*(m-beta*x)^2, +-a*rho/(4beta) <= R*(x) <= 0. +``` + +Since |h_l-h*_l|<=1/2, compare increments on [x,x+v]. Holes below both endpoints give the same increment v under both locations, and holes above both give zero. Only O(beta*v+1) holes whose ideal or actual locations meet the interval can contribute; each contributes an error at most 1/2. Consequently, for 0<=x<=x+v<=t, + +``` +|[R-R*](x+v)-[R-R*](x)| <= a*theta*v + O(a*rho), +|R(x+v)-R(x)| <= a*theta*v + K, +K <= 4a/theta (01. It is a norm lemma, not a complete fast inverse algorithm. + +Let s>=5, s=1. The exact periodic normalized Gaussian matrix is + +``` +N_(i,j mod s) = sum_(k congruent j-i mod s) + exp[-a*((beta_i+rho*k)^2-beta_(i+k)^2)]. +``` + +Its principal k=0 term is one. Let L be its k=-1 weighted cyclic shift and U its k=+1 shift. Thus + +``` +L_i = exp[-a*((beta_i-rho)^2-beta_(i-1)^2)], +U_i = exp[-a*((beta_i+rho)^2-beta_(i+1)^2)]. +``` + +All other k terms form R. The aliases k=±s and beyond remain in R; they are not discarded. + +For h consecutive lower factors the exponent telescopes to + +``` +beta_i^2-beta_(i-h)^2 - 2rho sum_(j=0..h-1) beta_(i-j) + h rho^2 + >= h rho theta - 1/4. +``` + +The upper factors obey the same bound. Therefore both alternating Neumann inverses exist, including over arbitrarily many cyclic wraps, and + +``` +||(I+L)^-1||, ||(I+U)^-1|| + <= B = exp(a/4)/(1-exp(-a rho theta)) + <= 2 exp(a/4)/theta. +``` + +The final inequality uses a>=1 and 0=2, + +``` +(beta_i+rho*k)^2-beta_(i+k)^2 + >= (rho*|k|-1/2)^2-1/4 + >= k^2-|k|. +``` + +Consequently + +``` +||R|| <= 2 exp(-2a)/(1-exp(-4a)) < 4 exp(-2a). +``` + +Write `A=(I+L)(I+U)`. Then N=A+(R-LU), and + +``` +||A^-1(R-LU)|| <= 5 B^2 exp(-2a) + <= 20 theta^-2 exp(-3a/2). +``` + +If `a>=4 ln(16/theta)`, this is bounded above by +`20 theta^4/16^6 < 1/2`. Hence + +``` +||N^-1|| <= 2 B^2 <= 8 exp(a/2)/theta^2. +``` + +Choose the integer `alpha=ceil(sqrt(4 ln(16/theta)/pi))`. With +`a0=4 ln(16/theta)`, the elementary ceiling inequality gives +`a<=2a0+2pi`; therefore + +``` +||N^-1|| <= 8 exp(pi) 16^4 theta^-6. +``` + +This is an explicit polynomial bound. For theta=Theta(1/d), alpha^2=O(log d) and the total original tensor scale has logarithm O(d log d)=o(p) for every fixed d=p^epsilon, epsilon<1. The period and aliases introduce no exception. + +## Exact controls + +`check_nearest_neighbor.py` checks 69 admissible finite (s,t) pairs and 93,976 product identities/bounds, including all skip gaps and two complete cyclic wraps. It uses exact rational exponents and no floating point. The exponential conclusions follow from monotonicity and the displayed convergent geometric sums. The output is `NEAREST-NEIGHBOR-CONTROLS.json`. + +## Remaining algorithmic obligation + +This lemma establishes that logarithmic alpha squared is analytically permitted. The old inverse implementation still incurs too much work at that alpha. A fast stable inverse, such as a fully normalized holes-completion algorithm, must still supply its actual movement, rounding and operation count. Erasing the inverse cost from the outer ledger would not follow from this norm proof alone. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md new file mode 100644 index 0000000000..876f44a23e --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md @@ -0,0 +1,60 @@ +# Partial tensor fusion: a strictly better prospective outer balance + +This is a prospective cost theorem, not an implemented resampler. It changes the full-fusion target to a smaller interface whose precision is payable. + +Suppose the normalized holes construction is completed so that a g-axis Gaussian group is a fixed number of tensor geometric/Toeplitz convolutions, diagonal factors, and corrections supported on the hole rows, and suppose its halo gather/scatter has the same asymptotic cost. Let d=p^epsilon and a=Theta(log d), theta=Theta(1/d). + +The exact normalizations developed in HOLES-AND-Q-LDU.md have local logarithmic range O(a/theta)=O(a d) per axis. Therefore g axes can be fused with O(p) guard bits if + +``` +g*a*d = O(p). +``` + +Taking `g=p^(1-epsilon-zeta)` for fixed zeta>0 pays the logarithm and leaves strict slack. There are d/g groups. A group convolution using the retained supplier with saving q costs, after division by full volume, + +``` +g^(1-q) polylog p. +``` + +Thus all Gaussian groups cost + +``` +(d/g)*g^(1-q) = d/g^q + = p^[epsilon-q*(1-epsilon-zeta)]. +``` + +Its saving is `(1+q)*(1-epsilon)-q*zeta`. The main synthetic transform, using deferred reservations, retains saving epsilon*q. Ignoring the arbitrarily small fixed zeta and strict absorption gaps, balancing yields + +``` +epsilon=(1+q)/(1+2q), +kappa=q*(1+q)/(1+2q). +``` + +This exceeds the old `q/(1+q)` by exactly + +``` +q^3 / ((1+q)*(1+2q)). +``` + +That is a new outer balance, rather than tighter rational margins in the old family. At the selected complex endpoint the difference is about 4.7e-9. For example take q equal to the selected complex saving minus 10^-10, zeta=10^-8, the displayed epsilon, and kappa ten^-11 below the smaller main/Gaussian margin. The resulting exact arithmetic target is approximately + +``` +0.00167266475042434534254271472, +``` + +about 4.55335e-9 above the selected 0.001672660197076918. This is only a prospective target until both missing interfaces below are established. + +## Other paid terms + +The bit routing saving is larger than q at the selected pair. Moving a group of O(g log p) small coordinate bits across spectator fields costs O(V g^(1-a_bit) polylog p) per group, hence O(V d/g^a_bit polylog p), below the convolution cost since a_bit>q. A superpolynomial untouched spectator suffix remains available while processing a group, so the compact supplier's descriptor work can still be paid; pointwise small-polynomial products require a separately paid small-field transpose but only logarithmic-width fields. + +The g-axis scalar normalization costs O(g a), and the stronger local weights cost O(g a d); both fit O(p) by construction. For local convolution window L=O(p/a+d), take a dyadic microtile N≫gL. The total halo volume is bounded by `(1+O(L/N))^g=O(1)`. The gauge increment bound implies total logarithmic variation O(g[a theta N+a/theta]); with N=Theta(gL), the extra condition is g^2<=O(d). It holds at the proposed epsilon near one, since 2(1-epsilon)q`, and requires `dK=o(p)`. Thus even if all Gaussian work disappeared, retaining that reservation argument would still force `epsilon<1/(1+q)`. It is insufficient to erase just the Gaussian row from a ledger. + +The selected outer logic was read in `research/complex-pr615-ceiling/vendor/multicold/engines/outer.py`; its seven margins and 47 slacks match the current 1094 assembly. The underlying detailed interfaces are in upstream sections 02, 04, 05, 06, 07, 08, 09, 10 and the cited `references/semantic-bulk/rad20/reports/` transfer notes. In particular, the current bulk Gaussian proof already uses local principal windows and short fields; proposing that same improvement again would not change the selected result. + +## 2. Deferred reservations: removing the power-width reservation condition + +This is an all-size **conditional transfer argument**, assuming the retained complete rectangular compact gadget, exact common-frame circuit, arbitrary coordinate router, and linear semantic guard. It does not require a new finite supplier identity. + +Let `G=4 ceil(log2 p)+6`, and choose `K=128G`. At a layer on `h` still-unprocessed selected bits, reserve enough complete K-bit chunks to provide the two front H-bit fields, one back H-bit field, and row bits, where `H=hG`. The number of reserved chunks is + +``` +R(h) = ceil(3hG/K) + O(log h). +``` + +For `h` above a fixed polynomial in `log p`, `R(h) <= h/16`. Select those chunks from the still-unprocessed axes. Apply the retained compact circuit to the other `h-R(h)` bits, borrowing all three dirty fields from the reserved subset. At the completed-call boundary it is exactly `C` on the active bits and identity on every reserved or spectator bit, for arbitrary initial coefficients and arbitrary dirty addresses. Therefore all reserved bits retain their original complete ranges and can be processed next. + +Repeat on the reserved subset. The processed subsets are disjoint. Stop once `h=polylog(p)` and apply those remaining kernels individually. This is **not** a recursive call on a truncated record or a clean-scratch claim: the entire original stream is still present, with the same full polynomial suffix, and every previously processed coordinate is an untouched spectator. Completed kernels on disjoint selected bits commute. All row padding is removed before moving to the next phase, so temporary factor-two padding never multiplies across phases. + +To establish the time bound, retain the original `m,W,s` recurrence and stop its internal recursion at `d^beta`. With `sigma=1-b`, any `lambda>max(1-a,sigma)` gives, with logarithmic factors suppressed, + +``` +F(h) <= O(h^lambda + h^sigma d^(beta*(1-sigma)) + 1). +``` + +This follows because a packed local basis change has width `O(hG)` and hence cost `O(V h^(1-a) polylog p)`. The term formerly written `(h d^c)^(1-a)` is now `h^(1-a) polylog p`. All compact-gadget exceptional fractions remain polynomially small with K=128G; the retained repair proof uses the current call's O(p)-bit address and is not replaced by a global exception count. + +Since `h_j <= d/16^j` above the polylogarithmic tail, + +``` +sum_j F(h_j) + = O((d^lambda + d^(sigma+beta*(1-sigma))) polylog p). +``` + +The extra physical rearrangements of complete existing fields cost +`O(V p^(1-a) polylog p)` by the retained arbitrary coordinate router, over O(log d) phases. Its polynomial descriptor work is paid by the unchanged superpolynomial polynomial-record suffix, at every internal call. This is additive, not free routing. + +Consequently, for every fixed `q < min(a,(1-beta)*b)`, the whole layer has + +``` +O(V [d^(1-q) + p^(1-a)] polylog p). +``` + +The common guard remains O(d): each completed phase changes semantic numerator/denominator bounds by O(h_j), and `sum h_j=O(d)`. Only the currently active phase contributes its internal excess. There is still exactly one truncation after the entire normalized layer. No power K=d^c and no condition c>q remains. K=o(ell) now follows from `ell=p^(1-epsilon)` for every fixed epsilon<1. + +The extra top round on the long axes can use the same layer after a paid coordinate rearrangement collecting its selected bits and K-bit complete neighborhoods. The common rounds use a single paid global layout router, rather than `p/K` separate K-swaps. Thus the transformed version of the synthetic cost is + +``` +O(V [p^(1-a) + ell*d^(1-q)] polylog p). +``` + +This is the strongest positive transfer established by the present note. It removes the geometric source of `q/(1+q)` in the compact compiler. It does not remove the independent O(Vd) Gaussian and scalar costs. + +### Checks + +`check_constructions.py` checks 1,000 exact reservation schedules and 5,460 entries of actual Gaussian-rational `C` tensor matrices. Every axis is processed once; the phasewise output equals the independently calculated closed tensor matrix. Omitting the deferred subset is a retained failing control. These checks support the written argument; they are not an all-size compiler implementation. + +## 3. Batched right-hand sides for the Gaussian boundary inverse + +This refinement applies to the precomputed noncyclic banded L/U factors in the retained principal-window solver. It uses the many spectator lines carrying the same axis matrix. + +For a triangular matrix with scalar bandwidth w, group consecutive rows into blocks of a power-of-two width `B in [w,2w)`. There is only one preceding block in a lower solve (one following block for an upper solve). Precompute each diagonal-block inverse. For B simultaneous right-hand sides, the update is + +``` +X_j = L_jj^-1 * (Y_j - L_j,j-1 * X_j-1). +``` + +Two B-by-B products implement each update. The seven-product identity gives `O(B^omega)` scalar operations, `omega=log_2(7)`. For z boundary rows and R>=B right-hand sides, splitting the columns into B-column batches gives + +``` +O(z R B^(omega-2)) +``` + +scalar work. Incomplete edge blocks and the last RHS batch are zero-padded only for the current product; this is a constant temporary volume increase, not a tensor-axis padding factor. The matrices may vary from row block to row block. No Toeplitz assumption is made. + +The original boundary has `z=O(M w/d + w)` rows in a length-M local window. Summed over axes and lines, the old regular contribution `V w^2` becomes + +``` +O(V w^(omega-1) polylog p). +``` + +The artificial-end contribution is handled by a polynomial-size microbox L; it is +`O(V d w^(omega-1)/L polylog p)`. The O(Vd polylog p) local field motion and interior-convolution cost is still paid and is still present. + +### Tape layout + +Expose the same axis and factor page used by the retained solver. Gather B adjacent RHS indices with a B-valued local field, moving that O(log p)-bit field across the current row field by the retained small-field transposition. It costs O(V log p) per axis. This charge belongs to the existing d-pass term. Read factor pages and RHS row blocks sequentially. Keep the immediately preceding solved B-by-B block on a fixed work tape. Products use a depth-first seven-product recursion, parking/restoring the other finitely many matrices at each node; scanning and quadrant rearrangement is O(B^2) per level and is dominated by B^omega. There is no random factor lookup and no separate tape for every RHS. + +### Precision + +Keep the retained O(p)-bit rounded factor entries and compute each B-by-B product exactly as a dyadic matrix product before one component rounding. An individual seven-product recursion has depth log B. Additions grow integers by only O(log B) bits at every product boundary, so exact products and their sums have O(p+log B)=O(p) bits. A block inverse is precomputed at the retained high precision, with its row-norm perturbation charged explicitly. + +Round after a completed block update, not between scalar products. The residual of the completed triangular system is bounded rowwise by `poly(p)*2^-P*(1+||X||)`, with P the retained large constant multiple of p. The retained inverse-norm bounds then give a uniform global solution error after absorption, independent of the number of row blocks. This is the same residual closure used by the original banded solver; it must be instantiated with new block-factor rounding constants in any adoption. An unproved claim of numerical stability of fast floating-point multiplication is unnecessary. + +For `u=p^r`, `w^2=O(p^(1-r))`. The revised regular boundary exponent is + +``` +((omega-1)/2)*(1-r), +``` + +whose saving at r=0 is `(3-log_2 7)/2 > 0.096`. Thus an implemented version could remove the old requirement r>kappa while keeping `epsilon+r<1`. It cannot remove the separate O(Vd) scan term by itself. + +### Exact checks and remaining check + +The script independently compares seven-product multiplication against classical multiplication, then compares 60 varied-size blocked triangular solves against exact Gaussian-elimination inverses (1,976 output entries). A dropped-predecessor-block control fails. This establishes finite algebraic orientation, padding and dependencies. A complete adaptation of the original numerical residual constants is still required before adopting the changed analytic interface. + +## 4. A concrete route toward tensor Gaussian fusion + +This section is **conjectural construction work**, not an established algorithm. The missing step is stated so it can be attacked. + +A possible way to avoid d separate Gaussian passes is to compute the tensor Gaussian on a microbox through one tensor convolution. For `rho=s/t`, a real Gaussian row has the exact scalar identity + +``` +exp[-a(j-rho*k)^2] + = exp[-a(1-rho)j^2] + exp[+a*rho(1-rho)k^2] + exp[-a*rho(j-k)^2]. +``` + +Integer translations of j and k add only linear factors, so each local rectangle is a diagonal, Toeplitz convolution, diagonal. Across axes, the diagonal factors fuse into one per-record multiplication. The central convolution is a tensor cyclic convolution after sufficient halos. + +With the current theta≈1/d, a local Gaussian halo has radius +`A=O(alpha*sqrt(p))`. To keep product halo volume constant, the core needs length `L=Omega(dA)`. But the fused diagonal logarithmic range is then +`O(d*theta*L^2/alpha^2)=O(d^2 p)`, not O(p). Thus the most obvious tensor-chirp fusion fails its precision ledger. This explicit failure should prevent a false positive. + +One changed hypothesis makes that favorable: choose source primes much closer to t, with `theta=Theta(d^-3)`. Then at `L=Theta(dA)`, that logarithmic range is O(p), and the product halo overhead is still constant. Such prime lengths are available eventually for every fixed epsilon<1: the explicit prime-counting error from Fiori–Kadiri–Swidinsky, Corollary 22, is `O(x sqrt(log x) exp(-c sqrt(log x)))`; on an interval of width `x/d^3`, with `log x=Theta(p^(1-epsilon))`, its error is o(x/(d^3 log x)). The interval contains far more than d primes, so distinct choices and `S/T=1-o(1)` follow. Trial division setup remains `n^o(1)`. This is an asymptotic existence argument with an enormous cutoff, not a practical prime-search claim. + +However two serious costs remain unproved: + +1. A power-of-two convolution window N has a useful core N-2A. Reinterpreting an axis by these variable non-dyadic cores is not the known binary-coordinate router. Padding every N-2A field to the next power of two costs 2^d. Performing the known small-field reshape one axis at a time costs Vd. A new simultaneous variable-core gather/scatter is needed, with full payload movement charged. +2. The inverse Gaussian correction is not itself one convolution. With theta=d^-3, the current phase-cell similarity weights range like `exp(O(u/theta))`, too large for its old O(p)-bit proof. The generic banded solve can avoid those enormous weights, but the present batched bound still has a d factor outside its favorable structured regime. The tensor inverse needs a different stable factorization or a sparse-boundary construction with a fully paid ledger. + +Therefore “choose closer primes” plus “use a multidimensional FFT” is not a proved bypass. It is an explicit construction program with two specified lemmas. + +## 5. Exact optimistic target and fully paid comparison + +Assume a new tensor-resampling interface gives + +``` +O(V [p^(1-a) + d^(1-q) + w^(omega-1)] polylog p) +``` + +for the complete expansion and inverse, including variable-core layout, coefficient factors, stable inverse, all restoration and only O(p)-bit records. Combine it with Section 2. The complete normalized saving is at least + +``` +min(a, epsilon*q, 1-epsilon*(1-q), + 1-((omega-1)/2)*(1-r), epsilon), +``` + +with strict `epsilon+r<1`, `q 0`, the complex endpoint exceeds q by `0.000000462676808103522925`, and the gamma gap is `0.000005`. The target exceeds the selected kappa by `0.000002319802923082` (about 0.1386894%). Letting epsilon approach one and q approach b would recover the complex saving itself, up to arbitrary strict slack. + +**Fully paid ledger today:** without the unproved tensor-resampling interface, the unchanged Vd term remains. It imposes `kappa<1-epsilon`, so maximizing the proven/reused terms still gives at most `b/(1+b)`. Sections 2 and 3 alone do not improve final kappa. Under the displayed sample, the paid Gaussian margin is only `0.00001`, so that sample FAILS the current interface. It is deliberately not emitted as a passing outer certificate. + +## 6. Reproduction and evidence + +Run from the parent workspace: + +``` +python3 work/radical_reduction_ultra/check_constructions.py +python3 work/radical_reduction_ultra/check_ledger.py +``` + +`CONTROLS.json` contains exact finite controls. `LEDGER.json` contains the optimistic row and the correctly rejected fully paid row. Neither imports a producer or mutates the source checkout. + +## Open next questions + +- Instantiate Section 2 against the exact current compact field-layout code and reject malformed reservoir choices. This is a bounded integration task; it could remove a real interface restriction even before a new final exponent is obtained. +- Carry the block residual argument in Section 3 through the retained P=32768p constants and factor-page format. The algebraic and layout reductions are explicit; numerical certification is the remaining bounded task. +- For the fusion program, first prove or disprove a simultaneous variable-core gather/scatter with O(V p^(1-a) polylog p) cost and no 2^d padding. This isolates the data-layout bottleneck before more analytic effort. +- Seek a stable tensor inverse for near-identity Gaussian resampling with theta≈d^-3 that avoids both enormous diagonal similarities and d sequential full scans. A finite small-d result is not enough; require the exact operator, record width and mass recurrence. + +## Primary sources and attribution + +- CrocSwap/integer-mult-bounds pinned upstream and transfer reports, in particular 02-streams, 04-swap, 05-layers, 06-transforms, 07-resampling, 08-assembly, 09-exact-arithmetic, 10-transposition; reviewed contributions by Douglas Colkitt and the cited independent contributors are retained unchanged. +- Harvey and van der Hoeven's original Gaussian resampling and multiplication interfaces are inherited through those source proofs. +- V. Strassen, *Gaussian Elimination is Not Optimal*, Numerische Mathematik 13 (1969), 354–356, DOI `10.1007/BF02165411`. The seven-product identity is reproduced and independently checked here; no modern matrix multiplication exponent is needed. +- A. Fiori, H. Kadiri, J. Swidinsky, *Sharper bounds for the error term in the prime number theorem*, arXiv:2206.12557v2, Corollary 22: https://arxiv.org/pdf/2206.12557 . Used only for the clearly marked close-prime construction program. + +## Later constructive developments + +The positive interpolation-tree and global two-tail results are in `STABLE-POSITIVE-TREE.md`, with exact controls in `POSITIVE-TREE-CONTROLS.json`. They provide bounded gauges for every signed-data intermediate and show that all high-level rank-one contributions collapse to two global running sums. Its Section 6 gives a normalized Sylvester factorization into a dirty Toeplitz shear, one well-conditioned full Toeplitz map, and one triangular Toeplitz map; `SYLVESTER-CONTROLS.json` checks that identity. + +The stronger fixed-width Gaussian norm theorem is independently audited in `FIXED-A-PERIODIC-NORM-AUDIT.md`, with `FIXED-A-LDU-CONTROLS.json`. It replaces the earlier sufficient logarithmic-width condition: for any fixed admissible a>=1, the periodic inverse has norm O(theta^-2). This is an analytic parameter improvement, not a completed fast inverse. + +The outer partial-fusion candidate in `PARTIAL-FUSION.md` remains prospective. Neither the stable, shallow in-place algebra for plain Toeplitz matrices nor bounded final Gaussian conditioning alone proves that the curved Gaussian coefficient gauges can be processed with only O(p) total guard bits when many axes are fused. That interface, and the actual halo movement cost, must be resolved before claiming a new kappa. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md new file mode 100644 index 0000000000..51920b3b76 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md @@ -0,0 +1,125 @@ +# A narrower positive route: fuse retained phase-cell interiors + +Status: a complete parameter ledger for a concrete conditional transfer. It avoids the unresolved normalized-erasure adapter. It still requires the new shallow in-place structured-matrix circuit to be integrated with the retained fast tensor supplier and a grouped sparse boundary gather. No new selected certificate is asserted. + +## Construction + +Keep the already reviewed phase-cell Schur decomposition, with + +``` +d=p^epsilon, u=alpha^2=Theta(p^r), w=Theta(p^((1-r)/2)), +0(1-r)/2. +``` + +The positive r may be arbitrarily small; it eventually dominates the old logarithmic lower bound on u. Thus this route can retain the established diagonal-dominance, rounded H, local inverse, and Schur residual proofs. The stronger fixed-a theorem is not needed here. + +Within each no-gap phase cell, the interior matrix is exactly + +``` +A=diag(a_j) G diag(a_j)^-1, +G_ij=G_(i-j), log(max a_j/min a_j)=O(u d). +``` + +The retained G is a symmetric truncated Gaussian Toeplitz matrix with `||G-I||<1/8`; hence G and G^-1 are uniformly conditioned. Apply its inverse by the new shallow in-place low-displacement circuit, rather than summing the two Gohberg-Semencul branches. The circuit's Toeplitz coefficients and Schur generators are precomputed. This is the algebraic connection to the new in-place work; the old source proof did not provide such a tensor-fusible circuit. + +Fuse + +``` +g=p^beta, beta=1-epsilon-r-zeta, +``` + +axes at a time, with fixed zeta>0. The total diagonal similarity guard is +`O(g u d)=O(p^(1-zeta))=o(p)`. Polylogarithmic circuit depth and bounded polynomial factor growth add only `O(g polylog p)=o(p)` bits. Every completed group has polynomial norm; error is allocated at the completed-group boundaries, preserving the retained O(p)-bit record model. + +### Dyadic interiors without exponential padding + +Inside a phase cell, repeatedly choose the largest power of two K such that `K+2w` fits the remaining interval, while its remaining length exceeds 4w. Make a piece of length K+2w, designate w endpoints on each side as boundary, and use its K remaining indices as the interior. The leftover interval of length at most 4w is all boundary. + +The remaining length at least halves on each step. Thus there are O(log d) pieces per original cell, every interior has dyadic length, and the total boundary size grows by only O(log d). No padding by a factor two per tensor axis is used. Schur fill still joins only the two w-sized endpoint groups of a piece, so its scalar bandwidth remains O(w). Distinct dyadic size classes require only O(log p) circuit schedules; inactive axes use identity stages. + +### Boundary arithmetic + +Let R denote the number of spectator right-hand sides, which eventually exceeds every polynomial in p. Batch them in groups of w. Every dense w-by-w block multiplication in the banded forward/backward solve, the border correction, and the boundary/interior couplings then costs O(w^omega) scalar products for w right-hand sides, omega=log_2(7)<2.8081. The retained no-pivot factors are precomputed; Strassen creates only polynomial intermediate growth in w, hence O(log p) extra guard bits. Use the existing global residual argument to bound the completed solve error. + +There are O(s*w*log(d)/d) boundary rows per axis. Consequently the total boundary arithmetic over all d axes is + +``` +O(V * w^(omega-1) * polylog(p)), +``` + +where V counts p-bit data volume. This includes H_BI and H_IB couplings; doing those couplings scalarly would incorrectly reintroduce V*w^2. + +### Five stages separate fused interiors from sparse corrections + +For the interior/boundary order write H=[A B; C D] and S=D-C A^-1 B. The exact identity + +``` +H^-1 = diag(A^-1,I) [I -B;0 I] diag(A,S^-1) + [I 0;-C I] diag(A^-1,I) +``` + +is verified by multiplication. Thus the only full-interior stages are A and A^-1. The two shears use original banded B and C, whose support is within O(w) of each artificial cell boundary. The middle factor splits into the commuting interior A and boundary S^-1. + +For a g-axis group, collect the union U of the O(w log d)-wide endpoint slabs. Its volume is O(V*g*w*log(d)/d). All sparse shear and boundary stages preserve this union; outside it they are identity. Process the g axis operators sequentially on U, without g scans of the full V data. Even a deliberately loose movement cost is + +``` +O(V*g^2*w*polylog(p)/d) per group, +O(V*g*w*polylog(p)) over all d/g groups. +``` + +Add one full scan per group to gather/scatter U, for O(V*d/g). This is below the proposed fused interior cost at the displayed parameters. Dense boundary arithmetic is still batched as above and costs O(V*w^(omega-1)*polylog p); scanning inactive entries in U is included in the movement bound. + +The union-slab gather/scatter and the evaluation of its piece catalogues must be implemented in the retained macro-routing model. The address fields have polynomial length and g*polylog(p)=o(p), but that observation alone is not a tape routing proof. Importantly, the cost budget now tolerates a sequential factor g on the **sparse union**, so the required route is weaker than a fully fused boundary solver. + +## Paid exponents + +Let q be strictly below the supplied complex saving and below the bit saving. The proposed dominant costs, normalized by V, are + +| Component | Power of p | Saving | +|---|---:|---:| +| Main transform with deferred reservations | 1-epsilon*q | epsilon*q | +| Fused Gaussian interior / convolution groups | epsilon-q*beta | 1-epsilon+q*beta | +| Group bit routing | epsilon-a_bit*beta | 1-epsilon+a_bit*beta | +| Sparse group scans | epsilon-beta | 1-epsilon+beta | +| Boundary algebra | (omega-1)*(1-r)/2 | 1-(omega-1)*(1-r)/2 | +| Sequential operations on sparse union | beta+(1-r)/2 | (1+r)/2-beta | + +Every entry has an explicit cost source above. The similarity/normalization condition is `epsilon+r+beta<1`. The old full-axis scan Vd is not part of the claimed implementation; until its replacement is proved, this remains conditional. + +Ignoring arbitrarily small zeta and strict absorption margins, balancing the first two rows gives + +``` +epsilon=(1+q-q*r)/(1+2q), +beta=(q-(1+q)*r)/(1+2q), +kappa=q*(1+q-q*r)/(1+2q). +``` + +This is feasible for `0 0. +``` + +Thus positive r is compatible with a strictly larger exponent, while retaining the old phase-cell analytic hypotheses. + +One explicit arithmetic target uses + +``` +q=0.001675462576808103522925, +r=0.000001, +zeta=0.00000001, +epsilon=(1+q-q*(r+zeta))/(1+2q), +beta=1-epsilon-r-zeta, +kappa=epsilon*q-0.00000000001. +``` + +It yields + +``` +epsilon = 0.9983301313358845139763404147924773... +beta = 0.0016688586641154860236595852075227... +kappa = 0.0016726647643531934852000135266363... +``` + +The proposed kappa exceeds the selected 0.001672660197076918 by about 4.5672763e-9. Its boundary saving is greater than 0.09595, its sparse-scan saving is greater than 0.0033387, its sparse-union saving exceeds 0.49833, and the bit saving is strictly larger than q. This target is deliberately **not** promoted: the new circuit/supplier integration and grouped sparse tape route must first be proved. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md new file mode 100644 index 0000000000..687c668a9d --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md @@ -0,0 +1,187 @@ +# Positive interpolation trees and two global inverse tails + +Status: exact algebra and an error bound for a compressed inverse representation. This note does not yet claim that the local tree operators admit the pure convolution factorization required for tensor fusion. + +## 1. Setup and signs + +Let `x_i=r^{q_i}`, `r=exp(2 a rho)>1`, and `q_i=floor(rho i+1/2)` for `0<=iu) x_j, +W_I(k)=d_(u-k)/M_I, 0<=k<|I|. +``` + +The equality `u-k=s-1-(k+c_I)` is the connection to the final reversed degree. + +Suppose `||y||_infinity<=1` and `||N0^-1||_infinity<=B`. Replace all leaves by their absolute values and call the resulting polynomial `R_I^+`. Completing the interval by the single outside monomial that chooses every node above I and no node below I gives + +``` +M_I * [z^k] R_I^+ <= [z^(k+c_I)] R_all^+ <= B d_(u-k). +``` + +Consequently + +``` +|[z^k] R_I| <= [z^k] R_I^+ <= B W_I(k). +``` + +This holds for every node of any contiguous binary interpolation tree, for arbitrary signed inputs. It does not require roots to remain correlated after rounding. In particular, it avoids the independently-rounded-erasure-quotient failure in ERASURE-ROUNDING-CONTROL.json. + +The merge identity is + +``` +R_(A union B)=F_B R_A + F_A R_B, +``` + +where A is the lower child and B the upper child. All fixed polynomial coefficients in this identity are positive. Each intermediate contribution, after multiplication by the chosen outside monomial, is a sub-sum of the final absolute Lagrange sum. Therefore bounded intermediate amplitude follows from the completed inverse bound, including in the presence of cancellation between input leaves. + +This is a height bound, not by itself a fast multiplication theorem. The gauges are degree dependent. Ordinary convolution of their unscaled coefficients can still require excessive precision. + +## 3. Partition bound for arbitrary selected nodes + +For any increasing nodes satisfying `x_(i+1)/x_i>=r`, let `e_k` be the kth elementary symmetric polynomial, and `M_k` the product of the k largest nodes. Comparing a selected k-subset to the k largest indices gives a partition of nonnegative index deficits. Hence + +``` +1 <= e_k/M_k <= C_r, +C_r=product_(h>=1)(1-r^-h)^-1. +``` + +For `r>=4`, `C_r<2`. This applies to every interval and to intervals with a node deleted. It controls the normalized fixed polynomial factors used by the tree. + +## 4. Two globally compatible far fields + +Write `E_k=e_k(x_0,...,x_(s-1))`, with E_0=1. Let `k=s-1-j`. + +If `i<=j-m`, then + +``` +0 <= 1-e_k(X without x_i)/E_k + = x_i e_(k-1)(X without x_i)/E_k + <= C_r x_i/x_(j+1) + <= C_r r^(-m-1). +``` + +The endpoint k=0 is exact and is read separately. If `i>=j+m`, then + +``` +e_k(X without x_i) + = E_(k+1)/x_i - e_(k+1)(X without x_i)/x_i, +``` + +and the second term relative to `E_(k+1)/x_i` is at most + +``` +C_r x_(j-1)/x_i <= C_r r^(-m-1). +``` + +The endpoint j=0 is again exact. Thus, after allowing a harmless factor `1/(1-C_r r^(-m-1))` to convert the relative approximation to a bound by the true positive entry, the two far fields are + +``` +(N0^-1 y)_j ~= (-1)^k [ + E_k/d_j * sum_(i<=j-m) w_i + +E_(k+1)/d_j * sum_(i>=j+m) w_i/x_i + + exact local strip |i-j|= (p log 2 + log(4 C_r B))/(log r) +``` + +makes the far-field approximation error at most `2^(-p-1)`. In the retained regime, `log B=O(log d)` and `log r=Theta(a)`, so `m=O(p/a+1)`. + +## 5. What must still be paid + +The construction reduces the data application problem to two globally compatible weighted running sums and a width-O(p/a) local strip. All coefficients, barycentric weights, and normalized tree factors may be precomputed: the source axis length is exp(O(p/d)), and any fixed polynomial power of it remains negligible compared with the full tensor volume. + +For a single axis, bounded positive interpolation trees are a plausible route to the local strip in `O(s p polylog p)` bit operations. To turn this into an exponent improvement, however, one must give a fixed sequence of convolution, diagonal, and permitted data-movement operations that can be fused over many axes. A merge written as a *sum* of two convolutions is not sufficient: tensoring it naively generates exponentially many branches. Neither the positive height bound nor the two-tail representation silently resolves that issue. + +## 6. A branch-free reduction of a normalized Sylvester merge + +This is an exact algebraic reduction, with a bounded-symbol condition. It reduces the remaining in-place question to one full Toeplitz map rather than a sum of two polynomial products. + +Take two children of size b. Choose an anchor zeta strictly between their node sets and rescale the variable. After swapping/scaling the child registers the merge has the form + +``` +S(B,A)=f(z) B(z)+z^b g(z^-1) A(z), +f(z)=product_(lower nodes x)(1+(x/zeta) z), +g(z^-1)=product_(upper nodes x)(1+(zeta/x) z^-1). +``` + +If zeta is the geometric mean of the largest lower node and smallest upper node, every displayed root parameter is at most `r^-1/2`. The sum of all parameters on either side is at most + +``` +eta = r^-1/2/(1-r^-1). +``` + +Thus the l1 norms of f and g are at most exp(eta), and the l1 norms of their reciprocal series are at most exp(eta/(1-r^-1/2)). All are constant and approach one as a increases. + +Let T_f be the 2b-by-2b lower triangular Toeplitz multiplication matrix for f. Put + +``` +h(z)=g(z^-1)/f(z), +K_ij=h_(i-b-j), +H_ij=h_(i-j), 0<=i,j>axis&1): + j=i|(1<4: + h=len(pending); r=(3*h*g+row_bits+k-1)//k + if r>=h or h-r<1:break + reserved=pending[:r];active=pending[r:] + assert r*k >= 3*h*g+row_bits + stages.append(dict(active=active,reserved=reserved)) + pending=reserved + stages.append(dict(active=pending,reserved=[])) + flat=[x for s in stages for x in s['active']] + assert sorted(flat)==list(range(d)) and len(set(flat))==d + return stages + +def run(): + global OPS + rng=random.Random(128731);mm=0;solve=0;rhs_entries=0 + for size in (1,2,4,8): + for _ in range(5): + A=[[F(rng.randint(-3,3),4) for _ in range(size)] for _ in range(size)] + B=[[F(rng.randint(-3,3),4) for _ in range(size)] for _ in range(size)] + before=OPS + assert mul(A,B)==classical(A,B) + assert OPS-before==7**(size.bit_length()-1) + mm+=1 + for width in (1,2,4,8): + for n in (1,3,7,10,17): + for k in (1,3,9): + A=[[F(0) for _ in range(n)] for _ in range(n)] + for i in range(n): + A[i][i]=F(1) + for j in range(max(0,i-width),i):A[i][j]=F(rng.randint(-2,2),8*width) + B=[[F(rng.randint(-4,4),8) for _ in range(k)] for _ in range(n)] + X=block_solve(A,B,width) + assert classical(A,X)==B + assert X==classical(inverse(A),B) + solve+=1;rhs_entries+=n*k + schedule_cases=0;exact_entries=0 + for d in range(1,1001): + s=reserve_schedule(d,128,1,8) + assert sum(len(t['active']) for t in s)==d + assert len(s)<=d + schedule_cases+=1 + for d in range(1,7): + s=reserve_schedule(d,8,1) + for col in range(1<>bit&1 else a) + assert v[row]==z;exact_entries+=1 + # Meaningful failure controls: the unresolved reserved coordinates matter. + s=reserve_schedule(6,8,1) + v=[ga(int(i==0)) for i in range(64)] + full=c_axes(v,range(6),6) + omitted=c_axes(v,s[0]['active'],6) + assert omitted!=full + # A wrong block dependency (drop the preceding solved block) fails. + A=[[F(1),F(0)],[F(1,4),F(1)]]; B=[[F(1)],[F(1)]] + assert B!=block_solve(A,B,1) + result=dict(status='PASS_EXACT_SCOPED_CONTROLS',strassen_cases=mm, + blocked_lower_solve_cases=solve,rhs_entries=rhs_entries, + reserve_schedules=schedule_cases,complex_tensor_matrix_entries=exact_entries, + negative_controls=['omit_reserved_axes','drop_preceding_solve_block'], + scalar_products_executed=OPS, + scope='Finite exact identities only; not a tape simulator or all-size compiler proof.') + print(json.dumps(result,indent=2)) + Path(__file__).with_name('CONTROLS.json').write_text(json.dumps(result,indent=2)+'\n') +if __name__=='__main__':run() diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_erasure_rounding.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_erasure_rounding.py new file mode 100644 index 0000000000..1f5f8e821a --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_erasure_rounding.py @@ -0,0 +1,36 @@ +#!/usr/bin/env python3 +"""Exact failure control for independent coefficient rounding before erasure division.""" +from fractions import Fraction as F +from pathlib import Path +import json + +def pmul(a,b): + c=[0]*(len(a)+len(b)-1) + for i,x in enumerate(a): + for j,y in enumerate(b):c[i+j]+=x*y + return c +s,t=29,32;H=t-s;rho=F(t,s) +q=[(2*t*i+s)//(2*s) for i in range(s)] +holes=[i for i in range(t) if i not in q] +assert holes==[5,16,27] +# exp(2*a*rho)=2^(2s); then natural source coefficient scale is 2^(-t*j*j). +base=2**(2*s);E=[1] +for h in holes:E=pmul(E,[-base**h,1]) +E0=abs(E[0]);P=1000;j=20 +# Relative 2^-P perturbation of Q_0. Exactly compute the formal quotient error. +delta=F(E0,2**P) +c=[F(0)]*(j+1) +for k in range(j+1): + rhs=delta if k==0 else F(0) + c[k]=(rhs-sum(E[h]*c[k-h] for h in range(1,min(H,k)+1)))/E[0] +error=abs(c[j])*(2**(t*j*j)) +min_error=F(2)**(t*j*j-2*s*holes[0]*j-P) +assert error>=min_error and min_error==2**6000 +assert delta/E0==F(1,2**P) +res=dict(status='PASS_ERASURE_ROUNDING_FAILURE_CONTROL',s=s,t=t,holes=holes, + relative_input_error='2^-1000',coefficient_index=j, + normalized_output_error_lower='2^6000', + exact_error_numerator_bits=error.numerator.bit_length(), + scope='Independent coefficient perturbation destroys divisibility by E. Not an impossibility result for correlated or exact-root-preserving algorithms.') +Path(__file__).with_name('ERASURE-ROUNDING-CONTROL.json').write_text(json.dumps(res,indent=2)+'\n') +print(json.dumps(res,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_fixed_a_ldu.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_fixed_a_ldu.py new file mode 100644 index 0000000000..ac7f96cd00 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_fixed_a_ldu.py @@ -0,0 +1,55 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from math import prod,ceil +from pathlib import Path +import json + +def p2(k):return F(2)**k +def mm(a,b):return [[sum(a[i][k]*b[k][j] for k in range(len(b))) for j in range(len(b))] for i in range(len(a))] +def symmetric(xs,n,complete): + p=[F(1)]+[F(0)]*n + for x in xs: + if complete: + for j in range(1,n+1):p[j]+=x*p[j-1] + else: + for j in range(n,0,-1):p[j]+=x*p[j-1] + return p[n] +def test(s,t,l,n): + ids=list(range(l,l+n));q={i:(2*t*i+s)//(2*s) for i in ids} + x={i:p2(2*s*t*q[i]) for i in ids} + e={i:p2(-t*t*i*i+(t*i-s*q[i])**2) for i in ids} + g=lambda i,j:p2(-(t*j-s*q[i])**2+(t*j-s*q[j])**2) + c={i:prod(1-x[k]/x[i] for k in ids if k=j else F(0)) + ur.append(e[j]/e[i]*symmetric([x[k] for k in ids if k<=i],j-i,True) if j>=i else F(0)) + if i>=j: + h=i-j + exponent=-(t*i-s*q[i])**2+(t*j-s*q[j])**2-t*t*h+2*t*sum(t*k-s*q[k] for k in ids if j=i: + h=j-i;z=(-1)**h*e[j]/e[i]*symmetric([x[k] for k in ids if k=j else F(0) for j in range(H)] for i in range(H)] + Ht=[[x[h]**j for h in holes] for j in range(H)] + D=[[1/peval(derivative,x[holes[i]]) if i==j else F(0) for j in range(H)] for i in range(H)] + A=[[Vi[t-1-j][h] for h in holes] for j in range(H)] + assert mm(mm(T,Ht),D)==A + # Complete a partial values vector to a degree0 for v in slacks.values()) +paid=dict(margins,unfused_gaussian=1-eps) +assert min(paid.values()) < k +assert 1-eps < selected +assert k>selected +# Require the missing map to be declared; calculations do not discharge it. +result=dict(status='PASS_OPTIMISTIC_LEDGER_AND_PAID_REJECTION', + selected_kappa=str(selected), target_kappa=str(k), + absolute_gain=str(k-selected), + proposed_parameters={name:str(v) for name,v in + dict(a=a,q=q,epsilon=eps,r=r,beta=beta).items()}, + omega_upper=str(omega_upper),omega_control='7^16 < 2^45', + optimistic_margins={name:str(v) for name,v in margins.items()}, + optimistic_slacks={name:str(v) for name,v in slacks.items()}, + fully_paid_margins={name:str(v) for name,v in paid.items()}, + fully_paid_accepts_target=False, + missing_lemma='Stable complete tensor Gaussian resampling including variable-core movement in stated cost and O(p) record width.', + scope='Arithmetic conditional target and explicit rejected existing-interface row; not a new multiplication certificate.') +Path(__file__).with_name('LEDGER.json').write_text(json.dumps(result,indent=2)+'\n') +print(json.dumps(result,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_nearest_neighbor.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_nearest_neighbor.py new file mode 100644 index 0000000000..d56f21f6b7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_nearest_neighbor.py @@ -0,0 +1,45 @@ +#!/usr/bin/env python3 +"""Exact exponent checks for the polynomial-condition nearest-neighbor lemma.""" +from fractions import Fraction as F +from pathlib import Path +import json +cases=products=0 +for t in range(6,29): + for s in range(max(5,(4*t+4)//5),t): + rho=F(t,s);theta=rho-1 + if theta>F(1,4):continue + q=[(2*t*j+s)//(2*s) for j in range(s)] + beta=[rho*j-q[j] for j in range(s)] + assert all(-F(1,2)<=v=h*rho*theta-F(1,4) + assert usum>=h*rho*theta-F(1,4) + products+=2 + cases+=1 +# With a >=4 log(16/theta), the proposed perturbation ratio is <=20 theta^4/16^6. +assert F(20,16**6)*F(1,4)**4 < F(1,2) +res=dict(status='PASS_EXACT_NEAREST_NEIGHBOR_EXPONENTS',parameter_pairs=cases, + cyclic_product_bounds=products,cycle_lengths_up_to=28, + controls=['all skip gaps checked','two full cyclic wraps checked'], + scope='Exact exponent identities and inequalities; real exponential conclusions follow from monotonicity and geometric sums.') +Path(__file__).with_name('NEAREST-NEIGHBOR-CONTROLS.json').write_text(json.dumps(res,indent=2)+'\n') +print(json.dumps(res,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_positive_tree.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_positive_tree.py new file mode 100644 index 0000000000..ebce2a90cf --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_positive_tree.py @@ -0,0 +1,106 @@ +#!/usr/bin/env python3 +"""Exact controls for the positive interpolation tree and two global tails.""" +from fractions import Fraction as F +from math import prod +from pathlib import Path +import json + +def mul(a,b): + c=[F(0)]*(len(a)+len(b)-1) + for i,x in enumerate(a): + for j,y in enumerate(b): c[i+j]+=x*y + return c + +def add(a,b): + c=[F(0)]*max(len(a),len(b)) + for i,x in enumerate(a): c[i]+=x + for i,x in enumerate(b): c[i]+=x + return c + +def factor(xs): + p=[F(1)] + for x in xs: p=mul(p,[F(1),x]) + return p + +def test(s,H,r): + t=s+H + qi=[(2*t*i+s)//(2*s) for i in range(s)] + xs=[F(r)**q for q in qi] + es=factor(xs) + ws=[] + for i in range(s): + den=prod(xs[i]-xs[j] for j in range(s) if j!=i) + ws.append(F((i*7+3)%11-5,den)) + whole=[F(0)]*s + abswhole=[F(0)]*s + for i in range(s): + e=factor(xs[:i]+xs[i+1:]) + whole=add(whole,[ws[i]*v for v in e]) + abswhole=add(abswhole,[abs(ws[i])*v for v in e]) + # Arbitrary positive target coefficient weights suffice for the exact + # completion inequality; these reproduce a Gaussian-shaped gauge. + ds=[F(1,r**(j*j)) for j in range(s)] + B=max(abswhole[s-1-j]/ds[j] for j in range(s)) + checked=0 + def tree(l,u): + nonlocal checked + if u-l==1: + fi=[F(1),xs[l]]; ri=[ws[l]]; ai=[abs(ws[l])] + else: + m=(l+u)//2 + fl,rl,al=tree(l,m); fr,rr,ar=tree(m,u) + fi=mul(fl,fr) + ri=add(mul(fr,rl),mul(fl,rr)) + ai=add(mul(fr,al),mul(fl,ar)) + Mh=prod(xs[u:]) + for k in range(u-l): + target=s-u+k + assert abs(ri[k])<=ai[k] + assert Mh*ai[k]<=abswhole[target] + assert ai[k]<=B*ds[u-1-k]/Mh + checked+=1 + return fi,ri,ai + ff,rr,aa=tree(0,s) + assert ff==es and rr==whole and aa==abswhole + left=factor(xs[:s//2]) + broken=[F(0)]*s + for i in range(s//2): + wrong=factor(xs[:i]+xs[i+1:s//2]) + broken=add(broken,[ws[i]*v for v in wrong]) + for i in range(s//2,s): + right=factor(xs[s//2:i]+xs[i+1:]) + broken=add(broken,[ws[i]*v for v in mul(left,right)]) + assert broken != whole + tails=0 + for j in range(s): + k=s-1-j + for i in range(s): + e=factor(xs[:i]+xs[i+1:])[k] + if i=e + assert (app-e)*r**(m+1)<=4*app + tails+=1 + if i>j: + m=i-j + app=es[k+1]/xs[i] + assert app>=e + assert (app-e)*r**(m+1)<=4*app + tails+=1 + return checked,tails + +n=nodes=tails=0 +for s in range(3,15): + for H in range(1,min(s,5)): + for r in (4,8): + c,d=test(s,H,r) + n+=1; nodes+=c; tails+=d +out={"status":"pass", "arithmetic":"exact rational", "instances":n, + "tree_coefficient_completion_inequalities":nodes, + "far_field_relative_error_inequalities":tails, + "negative_controls":["Omitting the upper-child factor changes the root polynomial.", + "The omitted upper-child factor is detected in every instance."], + "claims_excluded":["Fast normalized convolution implementation", "Tensor fusion", "New final exponent"]} +Path(__file__).with_name('POSITIVE-TREE-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n') +print(json.dumps(out,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_q_ldu.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_q_ldu.py new file mode 100644 index 0000000000..63f7d6eab0 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/check_q_ldu.py @@ -0,0 +1,64 @@ +#!/usr/bin/env python3 +"""Exact rational controls for Gaussian Toeplitz q-LDU and shifted exponents.""" +from fractions import Fraction as F +from pathlib import Path +import json + +def mm(a,b):return [[sum(x*y for x,y in zip(r,c)) for c in zip(*b)] for r in a] +def tr(a):return list(map(list,zip(*a))) +def ident(n):return [[F(i==j) for j in range(n)] for i in range(n)] +def powq(q,n):return q**n + +def factors(q,n): + Q=q*q;P=[F(1)] + for j in range(1,n):P.append(P[-1]*(1-Q**j)) + bc=lambda i,j:P[i]/(P[j]*P[i-j]) + L=[[q**((i-j)**2)*bc(i,j) if i>=j else F(0) for j in range(n)] for i in range(n)] + Li=[[(-q)**(i-j)*bc(i,j) if i>=j else F(0) for j in range(n)] for i in range(n)] + D=[[P[i] if i==j else F(0) for j in range(n)] for i in range(n)] + return L,Li,D + +cases=0;entries=0 +for q in (F(1,2),F(1,3),F(2,3)): + for n in range(1,11): + L,Li,D=factors(q,n) + G=[[q**((i-j)**2) for j in range(n)] for i in range(n)] + assert mm(mm(L,D),tr(L))==G + assert mm(L,Li)==ident(n) and mm(Li,L)==ident(n) + # Integer-power diagonal conjugation is an exact finite surrogate. + diagonal=[q**(i*i-3*i) for i in range(n)] + Lt=[[L[i][j]*diagonal[i]/diagonal[j] for j in range(n)] for i in range(n)] + Lit=[[Li[i][j]*diagonal[i]/diagonal[j] for j in range(n)] for i in range(n)] + assert mm(Lt,Lit)==ident(n) + cases+=1;entries+=n*n + +exponent_cases=0 +for rho in (F(1001,1000),F(17,16),F(3,2)): + theta=rho-1 + for c in (-2,0,3): + for i in range(15): + for j in range(i+1): + h=i-j;beta=theta*i-c + # D_i/D_j times q^h for the inverse L factor. + raw=theta*(i*i-j*j)-2*c*(i-j)-rho*h + condensed=(2*beta-rho)*h-theta*h*h + assert raw==condensed + exponent_cases+=1 + +# Rational validation of the perturbation bound in the worked 24x24 witness. +rbound=F(24,2**23);linv_bound=F(24) +error=linv_bound**2*rbound/(1-linv_bound*rbound) +assert linv_bound*rbound<1 and errora. These are the actual strict rate relations used by the +support-local compiler and its stopped recursion, not an assumption that +all bit operations have the complex supplier's exact endpoint. + +## Functional interface + +The reviewed outer proof consumes a globally complete dyadic tensor call +which applies normalized H0 simultaneously on the selected active bits, +acts as identity on arbitrary dirty helper/coarse/spectator coordinates at +completed-call boundaries, accepts O(p)-bit coefficient encodings, and costs +O(V g^(1-q) polylog p). The supply is used as a completed operator, with its +own scalar phases and working storage included. Outer Gaussian blocks do +not directly inspect the local donor edits, per-rank multiplicities, owner +replica count, or a particular parent-word ordering. + +The T74 source retains the same local normalized target, common-label and +mutable-source contract, whole-call hosting, exact wrappers/phases, and +restoration interface. Its new parent uses m=73 as before, with maximum +residual rank48 and fixed finite W=66304. Replacing the old finite stock W +by this fixed W changes constants and the O(log g) product-row reservation, +not any exponent in the support-local argument. The local word has480 +helpers, also unchanged in its interface. All donor reads are intended to +preserve the exact full local endpoint; our separate rational rebuild check +confirmed that algebra. The complete unmodified native/frame/phase replay has +now passed and certifies the recorded normalized implementation of those endpoints. +The source pin and hashes of the native, strict, control, and physical-invoice +receipts are bound in `NATIVE-ADMISSION-BINDING.json`. + +The larger fixed prefix allowance (2048 bits in the T74 global invoice) +also changes a constant in the linear semantic guard. The capped proof allows +a fixed supplier-dependent constant in O(g), and later O(g*a*B), with an O(p) +record constant chosen accordingly. It does not require the former numeric +prefix constant to remain512 or1024. The strict power guards B+X+r<1 and +E+r<1 remain in force and were checked independently. + +The neighborhood exposure still involves O(g log p) actual selected bits; +the T74 finite word neither requests ambient-p address permutations nor +changes the complete-field requirement. The retained bit supplier supplies +the same arbitrary-width equal-chunk interchange interface with more rate +than the chosen a. Source and sink scalings converting H0 to the raw H used +by the tensor compiler remain O(g) guard terms. There is no new finite-field +bridge or a new assumption that per-microbox coarse labels are complete. + +Thus the substitution does not require a new analytic lemma, removal of a +charged row, or a new kind of workspace. The updated finite supplier is now +verified by `work/target002_upstream/full-1661-verification/RESULT.json`, whose +status is `PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION`. The binding includes +the complete source manifest and actual fresh native, strict-assembly, physical +invoice, and control receipt hashes. The two rates above are completed conditional +substitutions: their remaining conditions are exactly the inherited all-size, +hosting, precision/tape, exact-recovery, and analytic hypotheses described above; +the matrix branch additionally requires omega_C<=9/4. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/NATIVE-ADMISSION-BINDING.json b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/NATIVE-ADMISSION-BINDING.json new file mode 100644 index 0000000000..72d30de3fd --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/NATIVE-ADMISSION-BINDING.json @@ -0,0 +1,155 @@ +{ + "status": "PASS_FRESH_NATIVE_ADMISSION_BOUND_TO_CONDITIONAL_SUBSTITUTION", + "scope": "Unmodified PR1661 complete fresh finite/native/global verifier passed; the independently reviewed phase-fusion recomposition retains all inherited all-size, hosting, precision/tape, exact-recovery and analytic hypotheses. The matrix corollary additionally assumes omega_C<=9/4.", + "source_pr": 1661, + "source_head": "f852af5e178ce035cd46181a29d9330bcbcaa206", + "source_manifest_sha256": "d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f", + "complex_lower_pinned": 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"PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION", + "control_counts": { + "admission": 114, + "bit": 12, + "controller": 15, + "independent_owner": 8, + "geometry": 6 + }, + "new_native_provider_checks": [ + { + "provider": "base", + "word_sha256": "0740b6f33927b9e60c9a7f35e7fcb2fd9ac6e2d3982112428b302a6966a18c73", + "native_steps": [ + { + "step": "native", + "exit_code": 0, + "seconds": 0.8915401669219136 + }, + { + "step": "ledgers", + "exit_code": 0, + "seconds": 0.6126613330561668 + }, + { + "step": "local-replay", + "exit_code": 0, + "seconds": 4.12045537494123 + }, + { + "step": "wrappers", + "exit_code": 0, + "seconds": 1.6204181250650436 + }, + { + "step": "phases", + "exit_code": 0, + "seconds": 105.53502279194072 + } + ] + }, + { + "provider": "extra", + "word_sha256": "17cc6c177c89eb264514e1efb3685ab9a621f3952a8f393a6a5eb28cf77e2884", + "native_steps": [ + { + "step": "native", + "exit_code": 0, + "seconds": 0.8948857497889549 + }, + { + "step": "ledgers", + "exit_code": 0, 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"d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f", + "target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/PROFILE.json": "f66dd80aff69525144eac18144c13d7d8be3d3c39028b3afa1b59a8edf53e8a3", + "target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/SPEC.json": "d6b6a2140f3a61ae8cfdd8b51a1131934d63d2dd8573c1b13eb2b9528f3c3db8" + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/REPRODUCE.txt b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/REPRODUCE.txt new file mode 100644 index 0000000000..42c6bb3033 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/REPRODUCE.txt @@ -0,0 +1,22 @@ +ARITHMETIC REPLAY (Python 3.11 or newer; standard library only) +From this phase_fusion_review directory: + python3 frontier_fusion/check_frontier_fusion.py + python3 frontier_fusion/independent_substitution_check.py + +These replay the rational native moment and both outer ledgers, including +negative controls. They do not rerun the entire native supplier. + +COMPLETE NATIVE REPLAY +Obtain CrocSwap/integer-mult-bounds at exact commit +f852af5e178ce035cd46181a29d9330bcbcaa206. Follow the requirements in +research/gmx-t74-p566-ep400/ and run its unmodified verify.py with a fresh +--work directory and --audit-shards 4. The retained RESULT.json and +NATIVE-ADMISSION-BINDING.json identify the completed fresh run and bind +its native, owner, phase, invoice, arithmetic and control receipts. +The full 121 MB long-precision assembly and its vendor tree are not duplicated +in this review bundle. Their hashes are retained; obtain/replay the pinned +source to reproduce those larger files. + +The two final exponents retain the inherited all-size hypotheses. The larger +value additionally assumes the supplied omega_C <= 9/4 theorem. Numerical and +finite native verification alone do not prove those external hypotheses. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex new file mode 100644 index 0000000000..78278aea07 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex @@ -0,0 +1,53 @@ +\section{Updated T74 supplier and final exponents}\label{sec:t74} + +The preceding proofs are parameterized by the effective supplier rate. We now substitute the independently replayed T74 complex supplier from \href{https://github.com/CrocSwap/integer-mult-bounds/pull/1661}{PR1661}, pinned at commit \texttt{f852af5e178ce035cd46181a29d9330bcbcaa206}. This is a new input to the same outer proof, with contributor attribution retained in the source package. The original numerical statements above remain valid special cases. + +Use the exact finite decimals +\[ + b_T=0.00167971337478,\qquad q_T=b_T-10^{-10}, + \qquad a_T=(q_T+b_T)/2. +\] +The independently retained bit rate $a_*=0.001733427039589844234749$ satisfies $q_T0 for x in assembly['outer'].values()) +checks=[] +for key,word in [('base','gauge44-word-566.json.gz'),('extra','extra-word-566.json.gz')]: + d=RUN/('native-new-'+key) + st=read(d/'STEPS.json') + assert [z['step'] for z in st]==['native','ledgers','local-replay','wrappers','phases'] + assert all(z['exit_code']==0 for z in st) + assert gzip.decompress((d/'local/FULL-WORD.json.gz').read_bytes())==gzip.decompress((SRC/'new'/word).read_bytes()) + checks.append({'provider':key,'word_sha256':sha(SRC/'new'/word),'native_steps':st}) +paths={RUN/'RESULT.json',RUN/'STAGES.json',RUN/'parent/RESULT.json',RUN/'strict/RESULT.json',RUN/'strict/STAGES.json',RUN/'strict/ASSEMBLY.json',SRC/'MANIFEST.json',SRC/'CLAIM.json',SRC/'new/SPEC.json',SRC/'new/PROFILE.json'} +for base in [RUN/'native-new-base',RUN/'native-new-extra',RUN/'strict']: + for f in base.rglob('*.json'): + if any(tag in f.name.upper() for tag in ['CONTROL','CHECK','REVIEW','AUDIT','REPLAY','INVOICE','STAGES']):paths.add(f) +for z in steps:paths.add(RUN/(z['name']+'.log')) +report={'status':'PASS_FRESH_NATIVE_ADMISSION_BOUND_TO_CONDITIONAL_SUBSTITUTION', + 'scope':'Unmodified PR1661 complete fresh finite/native/global verifier passed; the independently reviewed phase-fusion recomposition retains all inherited all-size, hosting, precision/tape, exact-recovery and analytic hypotheses. The matrix corollary additionally assumes omega_C<=9/4.', + 'source_pr':1661,'source_head':'f852af5e178ce035cd46181a29d9330bcbcaa206','source_manifest_sha256':r['manifest_sha256'], + 'complex_lower_pinned':assembly['complex_lower'],'complex_lower_used':'0.00167971337478','native_profile':assembly['complex_profile'], + 'full_status':r['status'],'control_counts':r['controls'],'new_native_provider_checks':checks, + 'claimed_upstream_kappa':r['kappa'],'recomposed_final_kappa':{'strassen':'0.001677160151714381','matrix_9_4':'0.001677600129305161'}, + 'receipt_sha256':{str(f.relative_to(W)):sha(f) for f in sorted(paths)}} +(P/'NATIVE-ADMISSION-BINDING.json').write_text(json.dumps(report,indent=2)+'\n') +print(report['status'],len(paths),'bound receipt files') diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/check_frontier_fusion.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/check_frontier_fusion.py new file mode 100644 index 0000000000..a8112e3cef --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/check_frontier_fusion.py @@ -0,0 +1,177 @@ +#!/usr/bin/env python3 +"""Recombine the pinned T74 supplier with the reviewed phase-fusion proof. + +This checks arithmetic and provenance, not the independent native replay. +The latter's completion is recorded separately; no all-size premise is waived. +""" +from fractions import Fraction as Q +from decimal import Decimal, localcontext +from pathlib import Path +from math import factorial +import hashlib, json + +HERE=Path(__file__).resolve().parent +WORK=HERE.parent +SPEC=WORK/'target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/SPEC.json' +CLAIM=WORK/'target002_upstream/pr1661/research/gmx-t74-p566-ep400/CLAIM.json' +spec=json.loads(SPEC.read_text()) +certified_complex=Q(spec['expected']['complex_lower']) +bit=Q('0.001733427039589844234749') +# Keep short independently checkable decimals strictly inside both endpoints. +b=Q('0.00167971337478') +q=b-Q(1,10**10) +a=(q+b)/2 +online=Q(spec['expected']['kappa']) +old=Q('0.001673360105405778') +assert b=2:x/=2;k+=1 + lo,hi=unit_log(x) + return down(lo+k*ln2[0]),up(hi+k*ln2[1]) +def exp(x): + assert 0<=x<1 + N=12 + lo=sum((x**j/factorial(j) for j in range(N)),Q(0)) + hi=lo+x**N/factorial(N)/(1-x/(N+1)) + return down(lo),up(hi) +def moment(endpoint): + lo=hi=Q(0) + for r,n in profile.items(): + l,h=log(Q(m,r)); weight=Q(n*r,m*W) + lo+=weight*exp(endpoint*l)[0] + hi+=weight*exp(endpoint*h)[1] + return down(lo),up(hi) + +native=moment(b) +assert native[1]<1 +assert moment(Q('0.001679713374785'))[0]>1 + +def ledger(kind, *, q_=q, zeta=Q(1,10**8), slack=Q(1,10**5), + old_scan=False, full_gauge=False, full_key=False): + r=Q(1,10**6); stop=Q(1,10**9) + if kind=='strassen': + c=Q(3,32) + eps=(1+q_+q_*(c-slack-r-zeta))/(1+q_)**2 + xi=eps*(1+q_)-c+slack + else: + c=Q(5,16) + xi=(Q(3,2)+q_-(1+q_)*(r+zeta)+slack)/(2+q_) + eps=(1+q_*(1-xi-r-zeta))/(1+q_) + beta=1-xi-r-zeta + saving=eps*q_ + # Deliberate strict 1e-11 gap after decimal truncation. + kappa=Q((saving*10**18).__floor__(),10**18)-Q(1,10**11) + # kappa is derived, not silently asserted to equal an expected decimal. + tau=1-a; sigma=1-b + internal=tau+(1-stop)*max(sigma-tau,Q(0)) + leaf=sigma+stop*(1-sigma) + lam=1-(q_+a)/2 + key=beta+1-eps + margins={ + 'main_transform':saving, + 'global_coordinate_router':a, + 'fused_gaussian':1-eps+q_*beta, + 'group_bit_router':1-eps+a*beta, + 'group_scans':1-eps+beta, + 'boundary_algebraic_matrix':c+xi-eps, + 'sparse_union_movement':1-eps-beta-Q(1,2)+xi, + 'polynomial_products':eps, + 'microbox_face_corrections':1-(eps+beta+3-8+2), + } + if kind=='strassen':margins['sparse_union_full_key']=margins['sparse_union_movement']-key + if old_scan:margins['old_per_axis_scan']=1-eps + if full_key and kind!='strassen':margins['unnecessary_full_key_sort']=margins['sparse_union_movement']-key + checks={ + 'actual_bit_above_effective':bit-a, + 'complex_above_effective':b-a, + 'q_positive':q_, 'complex_endpoint':b-q_, + 'q_below_effective':a-q_, 'stop_positive':stop, + 'stop_below_one':1-stop, + 'leaf_saving_above_effective':(1-stop)*b-a, + 'q_below_internal':1-internal-q_, 'q_below_leaf':1-leaf-q_, + 'lambda_above_tau':lam-tau, 'lambda_above_sigma':lam-sigma, + 'lambda_above_internal':lam-internal, + 'lambda_prime_above_lambda':1-q_-lam, + 'lambda_prime_above_leaf':1-q_-leaf, + 'epsilon_positive':eps,'epsilon_below_one':1-eps, + 'cap_above_band':xi-Q(1,2),'cap_below_cell':eps-xi, + 'group_positive':beta,'group_below_dimension':eps-beta, + 'support_router_span':1-beta, + 'group_gauge_power_slack':1-xi-r-beta, + 'actual_logarithmic_group_guard':1-xi-beta, + 'global_gamma_power_slack':1-eps-r, + 'actual_logarithmic_gamma_guard':1-eps, + 'alpha_logarithm_absorption':r, + 'group_key_fits_record':1-key, + 'membership_arithmetic_fits_record':1-beta-2*(1-eps), + 'microbox_halo_volume':8-3-beta, + 'positive_layout_slack':slack, + 'short_record_fallback':eps-a, + 'improvement_over_online_final':kappa-online, + 'improvement_over_previous_local_final':kappa-old, + 'complex_below_one_over_32':Q(1,32)-b, + } + if kind!='strassen': + checks['sparse_union_fraction_decays']=xi-beta-Q(1,2) + checks['matrix_exponent_above_supplied_bound']=Q(19,8)-Q(9,4) + checks['matrix_exponent_below_five_halves']=Q(5,2)-Q(19,8) + if full_gauge:checks['unrecentered_group_gauge']=1-eps-r-beta + checks.update({n+'_above_kappa':v-kappa for n,v in margins.items()}) + assert margins['main_transform']==margins['fused_gaussian'] + if kind=='strassen':assert margins['boundary_algebraic_matrix']==saving+slack + else:assert margins['sparse_union_movement']==saving+slack + return {'kind':kind,'kappa':kappa,'epsilon':eps,'xi':xi,'beta':beta, + 'q':q_,'a':a,'r0':r,'zeta':zeta,'slack':slack, + 'margins':margins,'strict_checks':checks} + +def encode(x): + if isinstance(x,Q): + with localcontext() as c: + c.prec=60 + return {'rational':str(x),'decimal':str(Decimal(x.numerator)/Decimal(x.denominator))} + if isinstance(x,dict):return {str(k):encode(v) for k,v in x.items()} + if isinstance(x,(list,tuple)):return [encode(v) for v in x] + return x + +rows={};controls={} +for kind in ['strassen','matrix_9_4']: + row=ledger(kind) + assert all(v>0 for v in row['strict_checks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + rows[kind]=row + variants=[('endpoint_equality',{'q_':b}),('zero_group_guard',{'zeta':Q(0)}), + ('zero_layout_slack',{'slack':Q(0)}),('old_per_axis_scan',{'old_scan':True}), + ('full_cell_gauge',{'full_gauge':True})] + if kind!='strassen':variants.append(('full_key_sort',{'full_key':True})) + controls[kind]={} + for name,kwargs in variants: + bad=ledger(kind,**kwargs) + failed={n:v for n,v in bad['strict_checks'].items() if v<=0} + assert failed,(kind,name) + controls[kind][name]=failed +out={'status':'PASS_EXACT_T74_PHASE_FUSION_ARITHMETIC', + 'scope':'Conditional recomposition of pinned supplier and already-reviewed outer proofs. Full fresh native replay is a separately tracked requirement; inherited all-size assumptions remain.', + 'complex_lower_short':b,'complex_lower_pinned':certified_complex, + 'native_moment':native,'profile':profile,'m':m,'W':W, + 'online_final_kappa':online,'rows':rows,'rejecting_controls':controls, + 'source_sha256':{str(p.relative_to(WORK)):hashlib.sha256(p.read_bytes()).hexdigest() for p in [SPEC,CLAIM]}} +(HERE/'FRONTIER-FUSION-ARITHMETIC.json').write_text(json.dumps(encode(out),indent=2)+'\n') +print(out['status']) +for name,row in rows.items(): + print(name,'FINAL KAPPA',encode(row['kappa'])['decimal'], + 'positive checks',len(row['strict_checks']), + 'gain over online',encode(row['kappa']-online)['decimal']) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/independent_substitution_check.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/independent_substitution_check.py new file mode 100644 index 0000000000..04ec764940 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/frontier_fusion/independent_substitution_check.py @@ -0,0 +1,45 @@ +"""Independent balance-equation check, not an execution of the proposed ledger.""" +from fractions import Fraction as F +from pathlib import Path +from decimal import Decimal,localcontext +import json,hashlib +D=Path(__file__).resolve().parent;W=D.parent +src=W/'target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/SPEC.json' +spec=json.loads(src.read_text());pinned=F(spec['expected']['complex_lower']) +b=F('0.00167971337478');q=b-F(1,10**10);eff=(q+b)/2;actual_bit=F('0.001733427039589844234749') +r=F(1,10**6);z=F(1,10**8);s=F(1,10**5) +assert 0eff +assert 7**16<2**45 +rows={} +for name,c,expected in [('strassen',F(3,32),F('0.001677160151714381')),('matrix_9_4',F(5,16),F('0.001677600129305161'))]: + if name=='strassen': + # Solve E(1+q)+q X=1+q(1-r-z), X-E(1+q)=s-c. + E=(1+q*(1-r-z+c-s))/(1+q)**2;X=E*(1+q)+s-c + else: + # Solve E(1+q)+q X=1+q(1-r-z), 2X-E(1+q)=1/2+s-r-z. + X=(F(3,2)+q+s-(1+q)*(r+z))/(2+q);E=(1+q*(1-r-z-X))/(1+q) + B=1-r-z-X + assert E*q==1-E+q*B + assert (c+X-E==E*q+s) if name=='strassen' else (1-E-B-F(1,2)+X==E*q+s) + assert F(1,2)0 and 1-E-r>0 + assert 1-B>0 and 1-B-2*(1-E)>0 + margins={'transform':E*q,'global_router':eff,'gaussian':1-E+q*B,'group_router':1-E+eff*B,'group_scans':1-E+B,'boundary':c+X-E,'sparse_union':1-E-B-F(1,2)+X,'polynomials':E,'microfaces':4-E-B} + if name=='strassen':margins['union_key_sort']=margins['sparse_union']-(B+1-E) + else:assert X-B-F(1,2)>0 + K=F((E*q*10**18).__floor__(),10**18)-F(1,10**11) + assert K==expected and min(margins.values())==E*q and all(v>K for v in margins.values()) + with localcontext()as cc:cc.prec=55 + rows[name]={'epsilon':str(E),'cap':str(X),'beta':str(B),'kappa':str(K),'margin_above_kappa':str(E*q-K),'checked_margin_count':len(margins)} +native_path=W/'target002_upstream/full-1661-verification/RESULT.json' +native_ok=False +if native_path.exists(): + native=json.loads(native_path.read_text()) + assert native['status']=='PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION' + assert F(native['kappa'])==F(spec['expected']['kappa']) + native_ok=True +report={'status':('PASS_INDEPENDENT_CONDITIONAL_SUBSTITUTION_REVIEW' if native_ok else 'PASS_INDEPENDENT_CONDITIONAL_SUBSTITUTION_REVIEW_PENDING_NATIVE'),'source_sha256':hashlib.sha256(src.read_bytes()).hexdigest(),'scope':('Independent balance derivation, margin comparisons, and functional-interface review; completed fresh finite/native admission is bound separately, with inherited all-size assumptions retained.' if native_ok else 'Independent balance derivation, margin comparisons, and functional-interface review; full fresh native admission remains pending.'),'rows':rows,'native_required_output':'target002_upstream/full-1661-verification/RESULT.json'} +(D/'INDEPENDENT-SUBSTITUTION-RECEIPT.json').write_text(json.dumps(report,indent=2)+'\n') +print(json.dumps(report,indent=2)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/gaussian_fixed_width_note.tex b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/gaussian_fixed_width_note.tex new file mode 100644 index 0000000000..e7a2c00d4e --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/gaussian_fixed_width_note.tex @@ -0,0 +1,212 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern,microtype,amsmath,amssymb,amsthm} +\usepackage[margin=1in]{geometry} +\usepackage[colorlinks=true,allcolors=blue]{hyperref} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert_\infty} +\newcommand{\Z}{\mathbb Z} +\title{Fixed-Width Gaussian Resampling Is Polynomially Conditioned} +\author{Research note prepared with OpenAI Codex} +\date{October 13, 2026} +\begin{document} +\maketitle +\begin{abstract} +An exact bilateral Newton factorization proves that the Gaussian +resampling inverse has norm $O(\theta^{-2})$ at a fixed Gaussian width, +where $\theta>0$ is the oversampling gap. This removes the width-growth +requirement from the conditioning argument. The proof uses convergent +infinite triangular factors and then periodizes them; it does not assert +operator-norm convergence of finite sections. A fast implementation and +its tensor cost remain separate obligations. No improved final +integer-multiplication exponent is claimed in this note. +\end{abstract} + +Let $1\le s5/6$. +At $a=1$, concavity gives +$1-e^{-\rho\theta}\ge1-e^{-\theta}\ge(1-e^{-1})\theta>3\theta/5$, +and $e^{1/2}<2$; substitution gives the stated constant $16$. +\end{proof} + +\begin{corollary}[The width may be fixed] +For the resampling convention $a=\pi\alpha^2$, one may take the fixed +integer $\alpha=2$ in the inverse-conditioning and normalization +argument. If all $d$ gaps satisfy $\theta_i\ge d^{-C}$ for a fixed $C$, +the total logarithmic normalization loss is $O(d\log d)$. +\end{corollary} +\begin{proof} +The theorem bounds each inverse by a fixed constant times $d^{2C}$. +The separate diagonal has norm at most $e^{a/4}$, a fixed constant. +Taking powers of two large enough to normalize these factors costs +$O(\log d)$ bits per axis; logarithms add under tensor products. +\end{proof} + +\paragraph{Scope and provenance.} +This strengthens the conditioning-only result in the earlier note +\emph{A Polynomial Condition Bound for Gaussian Resampling}. The +matrix is the one in Section~7 of the +\href{https://github.com/CrocSwap/integer-mult-bounds/tree/e6f9716621f4e01fc190617934cdfaf072bd976f}{inspected repository snapshot}. +Classical Newton interpolation and symmetric-polynomial identities are +used explicitly above; no generic-rank assumption is used. Two independent +agents checked the limiting and periodization arguments. Exact rational +controls cover negative and zero origins, finite $LDU$, inverse entries, +origin shifts and simultaneous periodicity. + +This theorem does not supply the new numerical implementation at these +parameters. Stable factor application, in-place tensor composition, +coefficient generation and all tape movement must still be priced. +The repository's inspected conditional exponent remains +$\kappa=0.001672660197076918$; no larger final exponent is certified here. +\end{document} diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/make_native_capsule.py b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/make_native_capsule.py new file mode 100644 index 0000000000..45a5aeb417 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/make_native_capsule.py @@ -0,0 +1,32 @@ +"""One-time capsule creation from the independently completed source-bound run.""" +from pathlib import Path +import argparse,json,hashlib,sys +sys.set_int_max_str_digits(500000) +P=Path(__file__).resolve().parent +ap=argparse.ArgumentParser(description=__doc__) +ap.add_argument('--native-source',type=Path,required=True) +ap.add_argument('--native-work',type=Path,required=True) +ap.add_argument('--output',type=Path,required=True) +a=ap.parse_args() +source=a.native_source.resolve();run=a.native_work.resolve() +b=json.loads((P/'frontier_fusion/NATIVE-ADMISSION-BINDING.json').read_text()) +def normalized(x): + if isinstance(x,dict):return {k:normalized(v)for k,v in x.items()if k not in {'seconds','log_sha256'}} + if isinstance(x,list): + y=[normalized(v)for v in x] + return sorted(y,key=lambda v:v['name']) if y and all(isinstance(v,dict)and'name'in v and'exit_code'in v for v in y) else y + if isinstance(x,str):return x.replace(str(run),'{native-work}').replace(str(source),'{native-source}') + return x +out={'schema':'phase-fusion-native-receipt-capsule/1','scope':'Canonical semantic hashes of the completed native JSON receipts. Elapsed seconds, stage-log digests, and the two root directory prefixes are normalized; concurrent named stage records are sorted. Actual current stage-log hashes are checked separately. Log byte hashes identify the historical run but are not compared across fresh runs.','source_commit':b['source_head'],'source_manifest_sha256':b['source_manifest_sha256'],'receipts':[]} +for rel,raw in b['receipt_sha256'].items(): + if rel.startswith('target002_upstream/full-1661-verification/'): + kind='native_work';r=rel.removeprefix('target002_upstream/full-1661-verification/');f=run/r + else: + prefix='target002_upstream/pr1661/research/gmx-t74-p566-ep400/' + assert rel.startswith(prefix);kind='native_source';r=rel.removeprefix(prefix);f=source/r + blob=f.read_bytes();assert hashlib.sha256(blob).hexdigest()==raw + row={'root':kind,'path':r,'historical_sha256':raw} + if f.suffix=='.json':row['semantic_sha256']=hashlib.sha256(json.dumps(normalized(json.loads(blob)),sort_keys=True,separators=(',',':')).encode()).hexdigest() + out['receipts'].append(row) +a.output.write_text(json.dumps(out,indent=2)+'\n') +print(len(out['receipts']),'receipts',sum('semantic_sha256'in r for r in out['receipts']),'canonical JSON hashes') diff --git a/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/phase_fusion_note.tex b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/phase_fusion_note.tex new file mode 100644 index 0000000000..4351a73450 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1673/capped-phase-fusion/phase_fusion_note.tex @@ -0,0 +1,460 @@ +\documentclass[11pt]{article} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,booktabs,longtable,array} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[colorlinks=true,linkcolor=blue,urlcolor=blue]{hyperref} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\newcommand{\norm}[1]{\left\lVert #1\right\rVert} +\newcommand{\diag}{\operatorname{diag}} +\newcommand{\poly}{\operatorname{poly}} +\newcommand{\plog}{\operatorname{polylog}} +\newcommand{\file}[1]{\href{#1}{\texttt{\detokenize{#1}}}} +\title{Capped phase-cell fusion in the conditional multiplication reduction} +\author{Research note prepared with OpenAI Codex assistance} +\date{13 October 2026} +\begin{document} +\maketitle + +\noindent\textbf{Review status: PASS\_CONDITIONAL\_PHASE\_FUSION\_REVIEW.} The capped local-gauge and layout changes have independent audits, followed by an integrated review of this note and its 44 exact strict slacks. The result below is conditional on the same finite suppliers and all-size computational interfaces as the retained selected result; it is not an unconditional integer-multiplication theorem and does not change the upstream selected certificate. + +\noindent\textbf{Updated final bounds.} The completely replayed T74 substitution in Section~\ref{sec:t74} gives $\kappa_T=0.001677160151714381$, or $\kappa_{T,M}=0.001677600129305161$ with the additional supplied matrix-exponent premise. Neither reaches the continuing target $0.002$. + +\section{Statement, inputs, and parameters} + +The retained selected saving is +\[ + \kappa_{\rm old}=0.001672660197076918. +\] +The retained independently replayed supplier lower bounds are +\[ + a_*=0.001733427039589844234749,\qquad + b_*=0.001675462676808103522925. +\] +Here $a_*$ is the bit saving and $b_*$ is the complex saving. They are supplied inputs, not recomputed finite constructions in this note. Put +\[ + q=b_*-10^{-10},\qquad a_{\rm eff}=(b_*+q)/2, + \qquad r_0=10^{-6},\quad \zeta=10^{-8},\quad s_0=10^{-5}, + \qquad c_0=3/32. +\] +The effective bit saving is deliberately weakened so that +$q2|t|+1$. Both nonzero parameters have controlled inverses, so zero shears cause no singular diagonal. Two- and three-$H$ identity words pad all permitted gate types to the same six-$H$ shape. Thus one original matching layer uses at most $(8m+2)$ simultaneous $H$ stages. + +For $g$ different axis circuits, Kronecker multiplication tensors the corresponding $H$ stages and multiplies the diagonal factors. There is no expansion into $2^g$ branches and no data padding. Coarse labels may select different switch and coefficient tables, while the stage schedule stays common. A tensor diagonal has coefficient $\prod_i d_i(x_i)$. In lexicographic order, cached prefix products are recomputed only on odometer carries; the total number of extra products is bounded by $L^g\sum_{k\geq1}k/L^k=O(L^g)$. Scanning the one-axis tables on a fixed number of tapes has the same amortized bound. Consequently each diagonal costs $O(V\plog p)$ rather than $O(Vg)$. + +Crucially, an $H$ stage is invoked \emph{globally across all complete coarse-label ranges}. A fixed coarse-label microbox does not contain complete high-coordinate bits to borrow. Each chosen local bit is placed in a complete $K$-bit neighborhood of its original axis, with +\[ + G=4\lceil\log_2p\rceil+6,\qquad K=128G. +\] +An individual neighborhood fits because $K+O(\log p)=o(\log s_i)$. The $gK$ selected positions can be exposed by a permutation supported on at most $2gK$ positions; the complement need not be stably compacted. Coarse labels are restored before the next diagonal is evaluated. + +The retained compact supplier's residual basis acts on its selected bits, not on the entire ambient address. Its control shuttles have width $O(hG)$ at a node with $h$ active bits. Reserve whole neighborhoods for the two front fields, one back field, and the full product row stock required by nested bit and complex calls. The completed active call is the desired kernel on active bits and identity on arbitrary dirty reserved coordinates. Thus these reserved selected bits can be deferred, rather than processed individually first. Their count is at most $h/16$ above a polylogarithmic tail. Padding adds whole rows only for the current phase and is deleted before the next one. + +The original sufficient wording $K/\log p\to\infty$ is replaced by the finite inequality actually used in its proof, $K\geq G+4\lceil\log_2p\rceil+10$. The present $K$ satisfies it. The support router implements all needed placements with support $O(h\plog p)$, and its short-record construction borrows only $\lceil\log_2p\rceil^2$ suffix bits. It does not move an entire long axis. + +Using supplier stopping exponent $10^{-9}$ gives a strict leaf saving above $a_{\rm eff}$ and $q$. The retained stopped recurrence and the geometric sum of deferred widths yield +\[ + O\bigl(Vg^{1-q}\plog p\bigr) +\] +for a complete global selected-bit call, including all exposure/restoration. The retained supplier implements $H_0=H/2$; multiplying by $2^g$ costs one shift scan and $O(g)$ guard bits. The full circuit therefore costs the same bound up to polylogarithms. The derivations and restrictions are in \file{TENSOR-COMPILER.md}, \file{SUPPLIER-SUPPORT-AUDIT.md}, and \file{SUPPORT-ROUTER.md}. + +\section{The two actual Gaussian operators} + +\subsection{Narrow compression and target-grid completion} +For one axis put $q_j=\lfloor\rho j+1/2\rfloor$ and $\beta_j=\rho j-q_j$. The retained narrow normalized Gaussian is $N=CTD$, where $C$ selects the $q_j$ target rows. Its off-diagonal row sum satisfies +\[ + \norm{N-I}_\infty\leq e^{-2a\theta}+5e^{-2a}. +\] +The two nearest directions must be bounded together; charging each by $e^{-2a\theta}$ would not give this result. With the chosen $a\geq4\log(16p)$ and $\theta\geq1/(4p)$, the row gap is at least $1/(8p)$ eventually. The globally fixed, truncated, structure-preserving rounded matrix $H$ can be chosen with $\norm{H-N}<1/(16p)$, diagonal exactly one, gap at least $1/(16p)$, row norm below two, and $\norm{H^{-1}}_\infty\leq16p$. Every local inverse uses a principal restriction of this same $H$. + +Let $E$ inject source coordinate $j$ at target coordinate $q_j$. On the full dyadic target grid define +\[ + H_{\rm aug}=EHC+(I-EC). +\] +It is permutation-similar to $H\oplus I$, so it inherits row dominance and inverse bounds. Its physical target bandwidth is at most $2w+2$, since $q$ increments by one or two. Its inverse on selected rows is $EH^{-1}C$. A diagonal normalization on selected rows, zero on holes, followed by the increasing $q$-image extraction gives exactly the retained compression approximation. + +\subsection{Broad expansion} +The broad forward matrix is distinct: +\[ + S_{kj}=\alpha^{-1}e^{-b_0(j-\lambda k)^2},\qquad + b_0=\pi/\alpha^2,\quad\lambda=1/\rho. +\] +Embed source values at the $q_j$ coordinates, zero elsewhere. Give every hole input column an identity column. The resulting square matrix $M$ agrees with $S$ on the intended zero-hole input. In selected/hole order, +\[ + M=\begin{bmatrix}\alpha^{-1}N_bD_b^{-1}&0\\ R&I\end{bmatrix}, + \quad b=\pi/(\alpha^2\rho^2),\quad + D_b=\diag(e^{b\beta_j^2}). +\] +The Gaussian row sum of $R$ is bounded. In particular $\norm{M}<3$ eventually; use $M/4$ as the completed forward map, making it contractive also on numerical errors in hole coordinates. Relative to the former $S/2$ normalization this contributes an extra $2^{-D}$ over $D$ axes. A single exact $2^D$ shift compensates it, increasing the retained scalar exponent $\Gamma$ by $D=o(p)$. + +On a phase cell $q_j=j+c$, a selected target block has entries +\[ + K_{ki}=\alpha^{-1}e^{-b_0(i-c-\lambda k)^2}. +\] +Its inverse is polynomially bounded even though an uncentered diagonal split can have a large range. The finite Gaussian LDU proof only uses gaps at least one and $|\beta|\leq1/2$. For +\[ + c_b=\prod_{m\geq1}(1-e^{-2b\rho m}) +\] +it gives +\[ + \norm{K^{-1}}_\infty + \leq\frac{\alpha e^{3b/4}c_b^{-4}} + {(1-e^{-b\rho\theta})^2}. +\] +Indeed $q'_k=k+\rho c$ has unit gaps and $\beta'_{q_j}=\beta_j$. Also +\[ + \log c_b^{-1} + =\sum_{h\geq1}\frac1{h(e^{2b\rho h}-1)} + \leq\frac{\pi^2}{12b\rho}, + \qquad c_b^{-4}\leq e^{\pi\alpha^2\rho/3}. +\] +Since $\alpha^2=O(\log p)$ and $\theta^{-1}=\poly(p)$, the bound is polynomial in $p$, uniformly over all contiguous cell subblocks. + +For a block anchor $m$, set $x=i-m$, $y=k-m$, and $\delta=(1-\lambda)m-c$, with $|\delta|\leq1/2$. The exact Toeplitz split is +\begin{align*} + K_{ki}&=D_{\rm row}(y)G(y,x)D_{\rm col}(x),\\ + G(y,x)&=e^{-b_0\lambda(x-y)^2},\\ + D_{\rm col}(x)&=e^{-b_0((1-\lambda)x^2+2\delta x)},\\ + D_{\rm row}(y)&=\alpha^{-1}e^{b_0\lambda(1-\lambda)y^2+ + 2b_0\delta\lambda y-b_0\delta^2}. +\end{align*} +The Gaussian Poisson formula bounds the symbol of $G$, with $\gamma=b_0\lambda$, between +\[ + \sqrt{\pi/\gamma}\,e^{-\pi^2/(4\gamma)} + \quad\hbox{and}\quad1+\sqrt{\pi/\gamma}. +\] +Thus every finite principal $G$ is SPD with polynomial condition number. Truncating at width $W$ and rounding symmetrically to sufficiently many $Cp$ bits preserve these bounds. The SPD shallow circuit applies. See \file{LOG-A-PHASE-SCHUR-AUDIT.md}, \file{BILATERAL-LDU.md}, and \file{FORWARD-SMALL-A-AUDIT.md} for the analytic details and exact local-identity controls. + +\section{Capping and recentering the phase interiors} + +All target holes and original physical period cuts are boundary coordinates. Selected target coordinates are consecutive inside an original phase cell, so no interior circuit is applied to a masked Toeplitz matrix. Use the maximum physical bandwidth $W$ for both directions. + +In each original $O(d)$-length cell, retain $W$ endpoints and repeatedly choose the largest dyadic interior of size at most $B$ that fits with a following $W$ separator. Once no full $B$ block remains, the dyadic-tail procedure halves the remaining length at every step until its size is $O(W)$, which is all boundary. There are $O(d/B+\log B)$ interiors and +\[ + O\bigl(W(d/B+\log B)\bigr) +\] +boundary coordinates per cell. Since $B=o(d)$, the boundary density is $O((W/B)\plog p)$. Each eliminated interior touches only its two neighboring $O(W)$ boundary groups. Schur fill therefore has half-bandwidth $O(W)$, regardless of the number of capped blocks. + +For the narrow exact cell weights +\[ + \omega_j=e^{-a(1/4-\beta_j^2)/\theta}, +\] +one step changes their logarithm by $2a\beta_j+a\theta=O(a)$. On a capped block use the exact rounded-weight ratio $\widehat\omega_j/\widehat\omega_m$ around one anchor. The common scalar cancels from the similarity, and its logarithmic range is $O(aB)$. The retained global rounding changes this by $O(1)$, because its $Cp$ precision dominates the original $O(ad)=o(p)$ range. This is symbolic recentering of the same matrix, not a new local rerounding. For the broad split, $|x|,|y|\leq B$ and $B\leq d$ give diagonal ranges +\[ + O\bigl((B+B^2/d)/\alpha^2\bigr)=O(B/\alpha^2)\subset O(aB). +\] +The number of blocks does not multiply circuit depth: they run as a direct sum on the $L$ existing coordinates. + +\section{Five factors, sparse windows, and batched boundaries} + +For either square operator write $Q=\left[\begin{smallmatrix}A&B_1\\C_1&D_1\end{smallmatrix}\right]$ in interior/boundary order, and $F=D_1-C_1A^{-1}B_1$. Exact multiplication gives +\begin{align} + Q^{-1}&=\diag(A^{-1},I)E_{-B_1}\diag(A,F^{-1}) + E_{-C_1}\diag(A^{-1},I),\\ + Q&=\diag(A,I)E_{C_1}\diag(A^{-1},F)E_{B_1}\diag(A,I). +\end{align} +The off-diagonal factors use the original banded $B_1,C_1$. The forward formula uses $F$ only as a matrix application; it does not require a forward Schur inverse or a forward LU-prefix assertion. For the narrow inverse, row diagonal dominance survives elimination. Its Schur row gap stays at least $1/\poly(p)$ and its row norm is bounded, so the retained no-pivot boundary solver applies. + +Batch independent right-hand sides in a dyadic width between $W$ and $2W$. A banded triangular solve becomes block updates +\[ + X_j=L_{jj}^{-1}(Y_j-L_{j,j-1}X_{j-1}). +\] +Precomputed diagonal inverses and two square block products implement each update. The seven-product recursion gives exponent $\omega=\log_2 7<45/16$, the latter inequality checked exactly by $7^{16}<2^{45}$. The same batching handles boundary/interior coupling products. Intermediate Strassen additions grow only polynomially in $W$. Round after a completed block update and bound its residual. Each local window has polynomial length $L$, the no-pivot factors and exact partial right-hand sides have polynomial bounds, and the completed system inverse has polynomial norm. Thus the uniform local residual is $\poly(p)2^{-P}$; there is no factor equal to the global number of windows. The total boundary cost is +\[ + O\bigl(V(d/B)W^{\omega-1}\plog p\bigr). +\] + +For a group let $J_i$ be its boundary set, and let $E_i$ also include every interior coordinate incident to $B_1$ or $C_1$. Then $|E_i|/L=O((W/B)\plog p)$ and each sparse factor fixes $E_i^c$. The union +\[ + U=\{x\in[L]^g:\exists i,\ x_i\in E_i\} +\] +has fraction $O(gW/B\plog p)$. It is closed under the sparse factors, including a possibly dense inverse on $J_i$. It is \emph{not} closed under a dense $A$ stage. Use two sparse windows, with scatter/regather around the intervening dense stage. + +For one sparse axis, $U_i=E_i\times[L]^{g-1}$ is a complete rectangle wholly present in $U$. Scan $U$, extract $U_i$, number $E_i$ in increasing order by a stream counter, transpose only this polynomial-size field, perform its boundary operation, and merge back. Only this one field is temporarily padded, and the padding is deleted before the next axis. Each sparse window costs $O(Vg^2W/B\plog p)$ per group. Across $d/g$ groups this is +\[ + O\bigl(V(dg/B)W\plog p\bigr). +\] +Every group also has a fixed number of full scans costing $O(V)$. Membership calculations use $O(g(\log s)^2\plog p)=o(p)$ bit operations per $p$-bit record. Detailed fixed-tape extraction and batching are in \file{PHASE-LAYOUT-AUDIT.md}. + +\section{Core faces, ordered embedding, and precision} + +The increasing map $j\mapsto q_j$ permits one product scan for a whole group. Enumerate the dyadic target rectangle. For target $k$, put +\[ + j=\max\{0,\lceil s(2k-1)/(2t)\rceil\}. +\] +It lies in the image exactly when $j0.000428$. The old core-face cost has exponent $\epsilon_M+\beta_M-3<-1.75$. + +The main and grouped Gaussian savings are equal, while the sparse-union saving has an exact $s_U$ surplus: +\begin{align*} + \epsilon_Mq&=1-\epsilon_M+q\beta_M,\\ + 1-\epsilon_M-\beta_M-1/2+\xi_M&=\epsilon_Mq+s_U. +\end{align*} +Their minimum equals +\[ + 0.001673360115405778453817047951720376\ldots>\kappa_M+10^{-11}. +\] +The new boundary saving is $5/16+\xi_M-\epsilon_M>0.0639$, so it is not binding. Global routing, group scans, packed products and core-face repairs have strictly larger savings. + +The optional full-key surcharge in the earlier conservative table is absent here: the actual sparse-union algorithm extracts complete rectangles $U_i$, transposes one field, and performs an ordered merge. It does not sort union records. In contrast, all sparse \emph{microhalo} radix keys remain paid in their negligible correction volume. Adding a full-key sort on the union is an explicit rejecting control; omitting it follows from the layout proof, not from the matrix exponent. The new parameter range and q-image ordering are independently checked in \file{MATRIX-LAYOUT-SIGNOFF.md}. + +The exact ledger \file{check_matrix_corollary.py} verifies 47 positive strict slacks and six rejecting controls, recorded in \file{MATRIX-COROLLARY-ARITHMETIC.json}. The gain over the upstream selected saving is $6.9990832886\times10^{-7}$. The gain over the separate Strassen result is $4.37755816907\times10^{-7}$. The strict minimum gap absorbs all remaining fixed logarithms as before. +\end{proof} + +\noindent\textbf{Corollary review status: PASS\_CONDITIONAL\_MATRIX\_PHASE\_FUSION\_REVIEW.} The additional matrix transfer, new-parameter layout and exact arithmetic passed their independent audits and integrated root review. Corollary~\ref{cor:matrix} retains its explicit additional matrix-exponent premise. The first theorem and its separate reviewed arithmetic receipt remain unchanged. + +\section{Updated T74 supplier and final exponents}\label{sec:t74} + +The preceding proofs are parameterized by the effective supplier rate. We now substitute the independently replayed T74 complex supplier from \href{https://github.com/CrocSwap/integer-mult-bounds/pull/1661}{PR1661}, pinned at commit \texttt{f852af5e178ce035cd46181a29d9330bcbcaa206}. This is a new input to the same outer proof, with contributor attribution retained in the source package. The original numerical statements above remain valid special cases. + +Use the exact finite decimals +\[ + b_T=0.00167971337478,\qquad q_T=b_T-10^{-10}, + \qquad a_T=(q_T+b_T)/2. +\] +The independently retained bit rate $a_*=0.001733427039589844234749$ satisfies $q_T0 for x in assembly['outer'].values()),'native strict inequalities failed') + words={} + for key,word in [('base','gauge44-word-566.json.gz'),('extra','extra-word-566.json.gz')]: + d=run/('native-new-'+key);st=read(d/'STEPS.json') + need([s['step'] for s in st]==['native','ledgers','local-replay','wrappers','phases'] and all(s['exit_code']==0 for s in st),'native word coverage failed: '+key) + need(gzip.decompress((d/'local/FULL-WORD.json.gz').read_bytes())==gzip.decompress((source/'new'/word).read_bytes()),'native word payload differs: '+key) + words[key]=sha(source/'new'/word) + caps=read(package/'frontier_fusion/NATIVE-ADMISSION-BINDING.json') + need(result['controls']==caps['control_counts'],'cold control coverage differs') + capsule=read(package/'NATIVE-RECEIPT-CAPSULE.json') + def native_normalized(x): + if isinstance(x,dict):return {k:native_normalized(v)for k,v in x.items()if k not in {'seconds','log_sha256'}} + if isinstance(x,list): + y=[native_normalized(v)for v in x] + return sorted(y,key=lambda v:v['name']) if y and all(isinstance(v,dict)and'name'in v and'exit_code'in v for v in y) else y + if isinstance(x,str):return x.replace(str(run),'{native-work}').replace(str(source),'{native-source}') + return x + bound={} + for z in capsule['receipts']: + root=run if z['root']=='native_work' else source + f=root/z['path'];need(f.is_file(),'missing native receipt: '+z['path']) + bound[z['root']+'/'+z['path']]=sha(f) + if 'semantic_sha256' in z: + digest=hashlib.sha256(json.dumps(native_normalized(read(f)),sort_keys=True,separators=(',',':')).encode()).hexdigest() + need(digest==z['semantic_sha256'],'native semantic receipt differs: '+z['path']) + files=[run/'RESULT.json',run/'STAGES.json',run/'strict/RESULT.json',run/'strict/ASSEMBLY.json'] + return {'status':'PASS_FRESH_NATIVE_RUN_BOUND_TO_PINNED_PHASE_FUSION','scope':'Existing complete cold native run checked against the pinned source and the independently replayed outer inputs; all-size assumptions remain.','source_commit':PIN,'source_manifest_sha256':sha(source/'MANIFEST.json'),'native_words':words,'complex_lower':assembly['complex_lower'],'profile':assembly['complex_profile'],'controls':result['controls'],'canonical_native_receipts_checked':46,'all_native_receipt_sha256':bound,'run_receipt_sha256':{str(f.relative_to(run)):sha(f) for f in files}} + +def main(): + ap=argparse.ArgumentParser();ap.add_argument('--package',type=Path,default=Path(__file__).resolve().parent);ap.add_argument('--native-source',type=Path);ap.add_argument('--native-work',type=Path);ap.add_argument('--output',type=Path);a=ap.parse_args() + need((a.native_source is None)==(a.native_work is None),'provide both --native-source and --native-work') + package=a.package.resolve() + if a.output:need(not a.output.resolve().is_relative_to(package),'--output must be outside the immutable package') + out=fast(package) + if a.native_source:out['cold_native_binding']=cold_binding(package,a.native_source,a.native_work) + if a.output:a.output.write_text(json.dumps(out,indent=2)+'\n') + print(out['status'],len(out['checks']),'exact control replays') + if 'cold_native_binding' in out:print(out['cold_native_binding']['status']) +if __name__=='__main__':main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr1786/FINE-PRIME-PADDING.md b/research/selective-borrowing-outer-retune/vendor/pr1786/FINE-PRIME-PADDING.md new file mode 100644 index 0000000000..bf9899ee21 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1786/FINE-PRIME-PADDING.md @@ -0,0 +1,355 @@ +# Fine prime padding with separate setup and online precision + +This extends the capped phase-cell construction of PR1673 and the +fractional-shift interiors and radix-union schedule of PR1758. The finite +supplier is the unchanged signed-554 T74 supplier admitted in PR1699. +The new mechanism is a finer choice of source primes. Its smaller phase +increment changes the chirp budget, provided coefficient setup is allowed +more precision than the online records. The boundary uses the retained +explicit Strassen format; no structured-solver stability claim is needed. + +The exact ledger gives conditional +`kappa=0.001679754788150048`, rounded strictly below the derived saving. +The conclusion retains the all-size hypotheses listed in section 8 below. +Finite checks do not prove those hypotheses, the cited prime theorem or the +complete multiplication theorem. No practical cutoff is claimed. + +## 1. A source-prime interval finer than 1/d + +Retain the dyadic target box of the assembly construction. Its dimension is +`d=Theta(p^epsilon)`, `0=229 that + + |pi(x)-li(x)| <= 0.2795 x (log x)^(-3/4) + exp(-sqrt(log x/6.455)). + +[Primary source, theorem 2](https://arxiv.org/html/1401.2689v2#S2). +This theorem is a cited mathematical input, not re-proved by the verifier. + +Apply it at the two endpoints of the requested interval about x=t_i. +The logarithmic-integral difference is at least `eta_p x/log x`. The sum +of the two errors divided by this main term is bounded, for a fixed C, by + + C p^2 (log x)^(1/4) exp(-c sqrt(log x)) -> 0. + +Indeed `log x=Theta(p^(1-epsilon))` and epsilon<1 is fixed, so its positive +power dominates log p. Consequently the interval contains eventually at +least `x/(8p^2 log x)` primes. This exceeds d, since x is exponential in +`p^(1-epsilon)` and the denominator and d are polynomial in p. Both target +sizes satisfy the estimate. Their intervals are disjoint, so all d source +primes can be made distinct, odd and coprime to their dyadic targets. + +The deterministic search used by the assembly still works: scan the +intervals and test candidates by trial division. Even scanning all O(r) +candidates with O(sqrt r) divisors and polynomial bit overhead costs +`O(d r^(3/2) poly(p))`. Its logarithm is `O(log p+p/d)=o(p)`, hence setup +is `n^o(1)=o(n)`. No unknown threshold is used as a claim of practicality. +The algorithm can fall back on the retained multiplier for smaller inputs. + +Let `rho_i=t_i/s_i=1+theta_i`. The interval gives + + 1/(4p^2) < theta_i <= 1/(2p^2-1) <= 1/p^2. + +Also `S/T > (1-1/(2p^2))^d >= 1-d/(2p^2) > 1/2` eventually, since +d<=p. Thus the same capacity bound `T/2=1/(4d)`. + +## 2. Polynomial conditioning survives the smaller increment + +Keep `alpha=ceil(sqrt((4/pi) log(16p)))` and `a=pi alpha^2`. Eventually +`4 log(16p)<=a<=5 log(16p)` and alpha^2=Theta(log p). The sharp row estimate +of PR1673, LOG-A-PHASE-SCHUR-AUDIT section 1, is valid for every theta>0: + + ||N-I||_infinity <= exp(-2 a theta)+5 exp(-2a). + +For theta>=1/(4p^2), its row gap is at least `1/(8p^2)` eventually; for +example `1-exp(-2a theta)>=1/(4p^2)` while the second term is O(p^-8). +Omitting the exponentially small nonzero diagonal aliases and setting the +diagonal exactly to one preserves a polynomial gap. Exponentially accurate +band truncation and coefficient rounding preserve half of it. Every principal +narrow block has the same row gap, and its Schur complements retain that gap. + +The broad bound in FORWARD-SMALL-A-AUDIT sections 1-3 already states its +dependence on theta explicitly: + + ||K^-1|| <= O(alpha^5 rho^2 theta^-2 exp(a rho/3)). + +This is polynomial for the new theta. Since rho=1+O(p^-2), the exponential +factor is at most p^2 eventually, theta^-2<=16p^4, and alpha^5 is +polylogarithmic. The finite and bilateral LDU entry bounds are geometric +in index distance and bound both row and column sums. The same estimates +apply to the finite contiguous sections used by elimination. The augmented +forward matrix adds identity hole columns and a Gaussian coupling of bounded +row and column sums. Thus a conservative common eventual bound p^10 covers +the relevant matrices and their inverse norms, in both induced row/column +norms and spectral norm. Polynomially small perturbations preserve it with +slightly larger constants. This bound is independent of the number of cells. + +Physical-prefix elimination still consists of one contiguous prefix and +disjoint future interiors separated by at least the bandwidth. The eliminated +matrix is a direct sum of such sections. A Schur inverse is a principal block +of the corresponding full inverse; its norm is therefore polynomial too. +Schur norms follow from `D-C A^-1 B`. These are the same prefix geometries +used in the retained dense no-pivot boundary argument, with new polynomial +constants. No general stability assertion for structured generators is used. + +## 3. Setup precision must increase; online precision need not + +The old absolute weight precision cannot be retained. For the narrow weights + + w_i=exp(-a(1/4-beta_i^2)/theta), + +the negative logarithm can now be as large as `a/(4 theta)=O(p^2 log p)`. +Rounding these weights absolutely to Cp bits can produce zero. The finite +checker includes a concrete underflow witness; this is a required change. + +Let `L_p=ceil(log2 p)` and let C be a sufficiently large fixed constant. +Use setup precision + + P_setup = C p + 64 p^2 L_p. + +Eventually `a<=16 L_p`, so `a/(4 theta)<=16 p^2 L_p`. Since log 2>1/2, +rounding a weight downward with absolute error at most `2^-P_setup` gives + + |w_hat_i-w_i|/w_i + <= 2^-P_setup exp(a/(4 theta)) + < 2^(-C p). + +The weights remain positive. Increasing C by a fixed amount supplies any +additional polynomial slack needed below. Computing these polynomially many +bits, exponentials, exact anchored ratios and local inverse/factor pages takes +polynomial time in p. Schoolbook arithmetic and ordinary convergent series +already suffice; no use of the improved multiplier at a larger precision is +required for this setup bound. + +Define one globally fixed rounded reference matrix H as in the retained +audit: within each original phase cell use positive rounded weights and a +single rounded, truncated Toeplitz kernel, and across cells use rounded +Gaussian entries, with diagonal exactly one. The weights use P_setup, while +the kernel and the actual online pages need only a sufficiently large +constant times p bits. For distances at most W, weight ratios have logarithmic +range O(aW)=o(p). The truncation tail is bounded directly in the Gaussian +entries, independently of the full-cell weight range. Hence H differs from +the exact Gaussian matrix by `poly(p) 2^(-(C-o(1))p)` in row and column norm. +This preserves the polynomial conditioning in section 2. All actual Schur +and inverse tables are computed for this H, with the same perturbation budget. + +The data circuit never multiplies by the full-cell weights. Use PR1758's +exact fractional-shift identities, whose algebra holds for every theta>0. +On a capped block of length at most B, the narrow remaining chirp has +logarithmic range O(a theta B^2); the broad chirp and its leftover linear +term have range O(b0 theta B^2+b0 theta B), where b0=pi/alpha^2. Each is at +most O(a B^2/p^2). The shifted Toeplitz kernels remain accretive: their shift +gap is at least theta/2, and the explicit symbol bound of PR1758 gives +polynomial inverse norms when theta^-1=O(p^2). Its accretive shallow-circuit +theorem therefore has polylog(p) guard bits per axis, as before. + +The ideal shifted representation and H need not have identical stored +coefficients. Their difference has the bound just established: replace the +old raw `Cp` weight precision by P_setup in the comparison in PR1758 section +2. Local shifted kernels and anchored chirps are generated directly, then +rounded at online precision. Across g axes the uncompleted amplification is + + exp(O(g a B^2/p^2 + g polylog p)). + +Thus O(p)-bit records suffice whenever + + beta + 2 xi - 2 < 1, beta < 1. + +The inverse comparison also stays within the error budget. If an ideal +interior A and the rounded reference H_II differ by Delta, then +`H_II^-1-A^-1=-H_II^-1 Delta A^-1`. Both inverse norms are polynomial; +the error changes only by a polynomial factor. Telescoping through the +constant number of completed factor stages, and through g tensor axes, +uses the stated chirp/prefix guard. Exact identity of their rounded +coefficient lists is neither assumed nor needed. + +Setup arithmetic at P_setup is separate from these records. It is not placed +inside an online motif call or multiplied into the per-record data cost. +PHASE-LAYOUT-AUDIT already permits polynomial coefficient/page generation +per complete microbox and axis, amortized by the superpolynomial inner and +spectator payload. Replacing its polynomial by another fixed polynomial +preserves that argument: `L^g` and every full unused source-axis range exceed +every fixed polynomial in p. The number of pages, capped pieces, axes and +required bits is polynomial. This is an asymptotic amortization, not a claim +that P_setup or the eventual cutoff is practical. + +## 4. Normalize completed maps with powers of two + +The exact Gaussian resampling identity and the coprime source/target +permutations do not require theta to be of order 1/d. They require the +positive separation and invertibility established above. The completed +forward, inverse and diagonal factors can have larger polynomial norms +than in the old parameter family. Divide each such completed map by an +additional dyadic factor of at most `2^ceil(100 log2 p)`. The conservative +p^10 bounds above are then below one. Restore the product of these exact +normalizers at the final scalar shift, together with the retained Gaussian +normalization. This preserves the exact transform identity. + +There are O(d) factors; the added shift has O(d log p)=o(p) bits because +epsilon<1. This also bounds the extra accuracy needed before exact recovery. +At a sufficiently large fixed C, data precision is `C p+o(p)=O(p)`. +Completed groups can be treated as contractions, while their uncompleted +prefixes/suffixes use the chirp and circuit guards of section 3. None of +these changes increases the semantic online dimension exponent C1=1. +Offline arithmetic is paid through the setup amortization, not that interface. +The normalizer is scalar: combine its powers into the group coefficients or +apply one scalar per completed group, rather than introducing a full scan per +axis. A constant number of these operations per group is absorbed by the +retained group-scan row. Global restoration is one already-paid exact shift. + +## 5. A larger but still polynomial locality window + +The old literal row-gap and locality constants must not be reused. A +conservative generic locality argument suffices. For a banded reference +matrix with `||H||_2,||H^-1||_2<=p^10`, put + + Q=I-p^-20 H* H. + +Its eigenvalues lie in `[0,1-p^-40]`. With `J=ceil(p^42)`, the polynomial +inverse `p^-20 sum_(j0`. +The locality polynomial is used in the proof and coefficient setup. The +online algorithm still uses the capped factorization and compiled circuits; +it does not perform J Neumann iterations on each data record. + +An original phase cell now has length Theta(p^2). Capped dyadic carving still +has boundary density `O((W/B) polylog p)`: the additional end-cell contribution +is `O(W log B/p^2)`, dominated when B<=p^2. Clipped physical period cuts and +holes remain boundary. Scalar Schur bandwidth stays O(W), since an eliminated +interior joins only its two neighboring O(W)-size separators. Larger cores +and halos change only fixed powers in offline setup and O(log p) address +fields. Their source axes are still superpolynomial in p. + +## 6. Retained Strassen boundary, circuit and union costs + +Use the explicit Strassen circuit, with `7^16<2^45`, so choose fixed matrix +target exponent `tau=45/16` and `c=(3-tau)/2=3/32`. PR1673's dense boundary +proof and fixed-format absolute-precision argument apply to the new polynomially +conditioned pages. Block scans, permutations, rewinds and linear combinations +remain charged. Boundary density is still W/B, so its cost power remains +`epsilon-xi+(tau-1)/2` and its saving is `c+xi-epsilon`. This applies to general +rounded blocks and needs no structured inverse algorithm. + +The interiors use PR1758's shifted Toeplitz representation and accretive +shallow-circuit theorem, with the smaller theta substituted in the explicit +symbol bound and with section 3's setup precision. Circuit algebra and depth +are unchanged; the condition guard remains polylog(p) per axis because all +inverse norms and theta^-1 are polynomial. + +Retain PR1758's radix-distributed union schedule. Complete lines stay in one +flag class and axis operators commute on distinct tensor factors. Its incidence +identity is `sum_r |S_r|=sum_x |F(x)|=sum_i |U_i|<=g mu V`. Each visit already +moves an O(p)-bit record; flag extraction and O(g log L)-bit headers are absorbed +when beta<1. We use PR1758's per-visit argument, not the obsolete 2 beta<1. +Union cost power remains `epsilon+1/2-xi`. No per-axis scan of the whole union +is added, and no vanishing-union fraction is assumed. Support routing, +whole-neighborhood reservation and interchange retain their stated interfaces. + +## 7. Explicit parameters and every paid row + +Use PR1699's certified endpoint +`b_cert=335950991225108904267/200000000000000000000000`. +Set `b=b_cert-10^-64`, `q=b-10^-64`, `a_eff=(q+b)/2`, +`r0=10^-64`, `zeta=10^-66`, layout slack `s=10^-64`, and stopped-recursion +parameter `stop=10^-70`. The bit endpoint is 0.001733427039589844234749. +Choose + + epsilon=9999999/10000000, + beta=9999997/10000000, + xi=epsilon(1+q)-3/32+s. + +Thus xi=0.90792965478815... . This is an explicit interior point; it does not +assert an attained limiting rate. Since beta0`. The global router also +exceeds the main saving. Every other row has larger slack. The minimum is +`epsilon q`, strictly above the rounded headline. + +Key side conditions are + + 1-beta=3 delta>0, + 1-beta-2(1-epsilon)=delta>0, + epsilon-beta=2 delta>0, + 1-epsilon-r0>0, + 1-(2xi-2)-r0-beta=0.184140990423...>0, + 1/20, 48>43. + +The ledger regenerates the remaining stopped-supplier, endpoint, logarithm +and layout conditions. A separate implementation reconstructs the point and +all nine powers. Fourteen rejected variants include the old theta guard, +old union scan, group=dimension, epsilon=1, an unsupported Strassen exponent +and halo/core choices outside the stated locality witness. These failures +are scoped to this proof and point, not universal lower bounds. + +## 8. Scope, sources and open obligations + +Newly supplied here: the effective fine-prime interval and capacity/search +argument; the polynomial setup-precision replacement; the theta-dependent +conditioning substitution; conservative locality/core choices; scalar +normalization accounting; and the exact parameter composition. + +Retained or cited mathematical inputs are Trudgian's prime theorem; PR1673's +Gaussian identities, bilateral/finite LDU, capped factorization, dense boundary +and layout arguments; PR1758's fractional-shift identities, accretive circuit +theorem and per-visit union schedule; and the admitted finite supplier. +Arithmetic and finite simulations do not prove these general statements. + +The multiplication conclusion remains conditional on the all-size compiler, +common labels, mutable sources, whole-call hosting, semantic online precision +C1=1, fixed tapes, prime-packing implementation, arbitrary-width equal-chunk +interchange, exact recovery and analytic transfer. The changed prime interval, +setup precision and locality scales are proved above rather than silently +substituted into old numeric hypotheses. No independent human review, +proof-assistant verification, practical cutoff or physical multiplication +implementation is claimed. + +Credit Vlado Vojdanovski for PR1673; IdeaOuter, as credited by PR1758, for +fractional-shift interiors and per-visit tag accounting; Jeremy Kothe and +contributors for the package series; and LeverRank for the second schedule +simulation. PR1699's notices retain the complete supplier lineage. PR1672 +published the signed-554 kernel first. New material here was prepared by +Chafik Boukhalfa with substantial OpenAI Codex assistance, under Apache-2.0. + +Open next questions: independently review setup versus online precision, +uniform prefix conditioning and the enlarged halo; formalize the fine-prime +transfer lemma; improve the finite supplier endpoint. Every new endpoint +still requires a complete paid native word and admission. The extreme +eventual threshold is intentionally not presented as a practical algorithm. diff --git a/research/selective-borrowing-outer-retune/vendor/pr1786/NOTICE.md b/research/selective-borrowing-outer-retune/vendor/pr1786/NOTICE.md new file mode 100644 index 0000000000..e491b2e587 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr1786/NOTICE.md @@ -0,0 +1,3 @@ +# PR1786 (chafreaky / Chafik Boukhalfa), fine source-prime padding + +`FINE-PRIME-PADDING.md` and `outer/check_fine_padding.py` (copied; only the first docstring line of the script is relabelled) are from CrocSwap/integer-mult-bounds PR1786, branch chafreaky/integer-mult-bounds `research/fine-prime-padding`, head `c52345caa318fc6b1b6e2659c57d746317a72823` (Apache-2.0, as stated in the PR). The mechanism (source primes at relative gap Theta(p^-2) from dyadic targets via Tim Trudgian's effective prime number theorem, arXiv:1401.2689v2 theorem 2; setup precision Cp + 64 p^2 ceil(log2 p); theta-dependent conditioning; polynomial halo p^48 and core p^128; dyadic normalizers; explicit Strassen boundary) is entirely theirs. This package changes only the explicit parameter point (epsilon, beta) and the surrounding stack, see `outer/FINE-PADDING-COMPOSITION.md`. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/FRONTIER.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/FRONTIER.json new file mode 100644 index 0000000000..bffa6d24cb --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/FRONTIER.json @@ -0,0 +1,118 @@ +{ + "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2", + "priority_revision": 4, + "initial_snapshot_utc": "2026-10-15T01:24:50.8637063Z", + "updated_since_utc": "2026-10-12T00:00:00.0000000Z", + "initial_prs_screened": 1874, + "initial_pages": 19, + "refresh_utc": "2026-10-15T02:15:55Z", + "highest_pr_number_in_refresh": 2251, + "updated_prs_in_refresh": 14, + "scope": "Complete recent title/body/head/status pagination and targeted relevant code review by four agents; not every discussion or diff.", + "maintainer_selected_kappa": "0.001672660197076918", + "highest_competing_conditional_open_claim_reviewed": { + "pr": 2220, + "kappa_approximately": "0.0016797549561255445" + }, + "new_kappa": "0.0016805", + "reviewed_refresh": [ + { + "number": 2251, + "title": "Research: exact 96-dimensional P104 shared correction space", + "updated_at": "2026-10-15T02:09:01Z", + "head": "822bceef780d8da9605798f0836ae5320e8e291e", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2251" + }, + { + "number": 2250, + "title": "Execute selected complex compact adapters with sequential repair", + "updated_at": "2026-10-15T02:06:11Z", + "head": "d404b48a1e7b147556b309497d55089c837dddb2", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2250" + }, + { + "number": 2249, + "title": "Research: high-arity connector toggle cost floor", + "updated_at": "2026-10-15T02:05:27Z", + "head": "0c6ed7958433b2cbc0beaa331ff882aa08221a0d", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2249" + }, + { + "number": 2248, + "title": "Research exact Q8 three-role target minima", + "updated_at": "2026-10-15T02:04:12Z", + "head": "835bd7b1877f56d6d9f4eecd7d1ee32f31593e1e", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2248" + }, + { + "number": 2247, + "title": "Add exact E6 sheared-frame cost census", + "updated_at": "2026-10-15T02:00:13Z", + "head": "24456c7c97752b4f301aa8b3873d3b20b7d1a9ae", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2247" + }, + { + "number": 2246, + "title": "Research: exact four-call floor for A5 conjugate 5-cycle pair (no new kappa)", + "updated_at": "2026-10-15T01:55:30Z", + "head": "0e2b2b201e277858fdc9bcb01d3373a73bbc9d7c", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2246" + }, + { + "number": 2244, + "title": "Paper-side line counts per dimension: Lagrangian bound v \u2264 T(2^k\u2032+1) with exact ladder D4/D6/E7/E8 and no-gos scoped to windows h \u2264 13 (no \u03ba claim)", + "updated_at": "2026-10-15T01:47:15Z", + "head": "c4505d1fe313510196147479938c0713e9acd98e", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2244" + }, + { + "number": 1673, + "title": "Conditional kappa = 0.001677160151714381 via capped phase-cell fusion", + "updated_at": "2026-10-15T01:41:33Z", + "head": "85ddaa312e463cfaca98d49a2a262994551cec4e", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1673" + }, + { + "number": 2239, + "title": "Tooling: 1.88\u00d7 faster production joint-dual replay via containment masks with identical receipts", + "updated_at": "2026-10-15T01:40:36Z", + "head": "8fdb992c7d40239f8a5b7602fa6f4636f3d97961", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2239" + }, + { + "number": 2237, + "title": "Research exact p=25 root-pair decoder boundary", + "updated_at": "2026-10-15T01:39:10Z", + "head": "bc63155a531d814dd8fa98993d84dc287756ad47", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2237" + }, + { + "number": 2241, + "title": "Research: temporary two-output sum anchor (local construction)", + "updated_at": "2026-10-15T01:38:44Z", + "head": "5688d8b17830dbb813fba5a509f52449383903fa", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2241" + }, + { + "number": 2240, + "title": "Research bounded E6 one-extra frame decoder screen", + "updated_at": "2026-10-15T01:37:13Z", + "head": "c8a2b37056e766715c457f56e2dd8172f5bdbccd", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2240" + }, + { + "number": 349, + "title": "Certify PR288's restricted sign-gauge obstruction with four constraints (no new \u03ba)", + "updated_at": "2026-10-15T01:28:33Z", + "head": "4b0e84065c2902808e33f14f423de43ed12b92a0", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/349" + }, + { + "number": 2236, + "title": "Research: 16-event seven-source connector boundary (no new kappa)", + "updated_at": "2026-10-15T01:26:39Z", + "head": "be03870d764e7875ea9a9f6cc69933c9b1a2456a", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2236" + } + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/LICENSE b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. 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We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/MANIFEST.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/MANIFEST.json new file 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"f1a3f817d9322fde668eb9c02ac60d1229e8680d8a2daab04b90c4d44438c34b", + "verify.py": "6811dd2e3b393ad14181f32556eaedaba45271759e3595e93f30edebac776a9a" + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/NOTICE.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/NOTICE.md new file mode 100644 index 0000000000..cf8d16b6fc --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/NOTICE.md @@ -0,0 +1,41 @@ +# Attribution and retained licenses + +Prepared for Dugongue with substantial OpenAI Codex assistance: one primary +agent and three construction, theory and independent-audit agents. Independent +agent checks are not independent human review. New work is Apache-2.0, under +the accompanying LICENSE. Required upstream notices are retained verbatim. + +Cinna Davis (1337 isnot 1337), PR 2226, commit +dacaf3d81f1936bb55b8454b1244e5d014272754, supplied the anticipatory dirty-reference +borrowing architecture, initial compensation and private cleanup. The bundled +PR2226 replay and public source word/certificate retain that provenance. This +package contributes the 44-selection, both full native words and reflected +orientation, complete normalized ABIs, actual T 8 owner repacking, finite invoice, +and the resulting improved conditional exponent; it does not claim invention +of the predecessor borrowing architecture. + +The exact Clifford compiler/native interpreter retain the PR1972/1991, +PR1862 and FRAME-SA504 notices in local/vendor. The local source word retains +the gauge44/leaver8, frame and selected-complex block lineage. PR1005's +research/gauge44-leaver8 supplies PLAN, canonical hosted parent and DATA-PHASES +lineage; its notices and Apache license are in owners/notices. The independent +raw parent has the distinct retained path +research/complex-k492-owner-repair/vendor/multicold/vendor/pr558/vendor/pr552/complex-global/vendor535/inputs/ROUND20-PARENT-TEMPLATE.json.gz. +Its SHA256 is025448abdf2012d327e50a009084e1501690d05e650300d2982836aa0350ae00. + +The arithmetic engines and retained bit EXPECTED data are copied byte-for-byte +from CrocSwap/integer-mult-bounds main0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; +arithmetic/SOURCES.json gives exact upstream paths/URLs/hashes. The retained +outer engine credits eumemic, icekylinx, Zhihao Chen/PR23, and hipotures/RaD for +the balanced transform. Its 47 constraints are preserved. The exact linear +semantic guard and near-parent recurrence follow the reviewed RaD transfer: +https://github.com/hipotures/rad/blob/45d9b60355872041f9275f77c057514def0d45bc/reports/review-semantic-child-guard.md +and review-whole-complex-transfer.md at that revision. These identify the +original CrocSwap manuscript bcd4ebde8692383539f8a48734e5fbf3a18a32c2 and retained +compact-control revision6e564879f51ae16f23d392e9e196c605f36d90df. + +No source imports the contributor's private Lean library. The eleven universal +Lean lemmas import only Mathlib; the Python replay uses bundled public inputs +and the standard library. Earlier personal multitape/returned-workspace work +informed decomposition, but no full Problem 1.8 theorem is claimed as ported, +imported or required here. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/PROOF-LOCAL.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/PROOF-LOCAL.md new file mode 100644 index 0000000000..d16c7549ce --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/PROOF-LOCAL.md @@ -0,0 +1,205 @@ +# Exact local construction and finite charges + +This campaign's positive local contribution is an actual two-orientation native +word for a **44-alias sub-witness of Cinna Davis's PR #2226**. It retains the 48 +high-helper aliases except the four from block 632. The predecessor's original +80-alias proposal, scalar architecture, initial compensation mechanism and +source attribution remain credited to PR #2226 and its retained lineage. The new +work here is explicit chronological emission, reflected completion, every native +operator/phase binding, actual endpoint normalizers, and source-bound finite +precision and work charges. These are finite arithmetic/operator certificates; +they are not new Lean multiplication theorems. + +The exact selected local profile is + +``` +{1:2424, 2:418, 3:429, 4:141, 5:42, 8:8} +``` + +Its mass is 5385, including COPY 64. There are 436 actual physical helpers with +residual-width inventory `{4:40, 5:4, 7:27, 8:125, 9:240}`. Width always means +`dim(final frame) - dim(initial frame)`; nonzero entrances are retained. The +660 independent public/data/helper inputs are arbitrary. Seventy-five additional +owned zero arrays are used by FULL-frame inverse cleanup, and one COPY array is +used sequentially. Two additional coefficient-work reserves are charged for +literal Horner arithmetic. These arrays are not declared free dirty storage. + +## Universal finite execution and dirty restoration + +Fix the explicit emitted forward and backward traces and the native recipes +named by their chronological occurrence references. On every assignment of the +660 public input arrays, with the declared private arrays initially zero: + +* The normalized forward endpoint is `Y_i <- Y_i + X_i`, with every other + public input restored. +* The normalized backward endpoint is `X_i <- X_i - Y_i`, with every other + public input restored. +* Every owned private array returns to zero. An explicit exact CLEAR postlude + is charged and replayed as well. It does not rely on an implicit zero return. + +The emitting replay and a separately written row interpreter both verify the +literal programs over exact rational coefficients and a free noncommutative +operator algebra. They replay the COPY guards, resets, changed references, all +cleanup operations and the fixed coefficient chronology. A normalized row has +an independent input label and an ordered word in the reference-change +operators; only adjacent inverse pairs cancel. Thus arbitrary dirty inputs and +operator order, rather than only the all-zero helper assignment, are checked. + +## The actual backward construction + +The forward word is extended by literal destructive COPY/CLEAR semantics; +private incoming columns are ignored and private output rows vanish. Reverse +the scalar chronology, transpose each ADD, retain each real SCALE coefficient, +and complement every frame. A COPY is reflected into an ADD to its source +followed by CLEAR; CLEAR remains CLEAR. Existing owned-zero arrays are explicitly +reframed only when their literal content is zero. Conjugating the source sign at +entry and exit supplies the negative shear. + +For a forward logical change `K_i`, the reflected logical change is + +``` +J_i = F K_i^dagger F^(-1). +``` + +Using `K_i^dagger` alone at complemented frames would be invalid: the resulting +handoffs can have positive Fourier rank. The actual `J_i` choices are compiled +and every reflected handoff is independently checked to have rank zero. +The complete backward endpoint is independently replayed, rather than inferred +from an inverse of a program whose private COPY operation is noninvertible. + +## Native operators and complete endpoint ABI + +Each literal transport is lowered through the retained exact Clifford compiler. +A separately implemented signed-Pauli interpreter verifies all eighteen Pauli +generators and an exact Gaussian state fixes global phase. This determines the +complete Clifford operator, not merely its binary symplectic matrix. There are +2,996 distinct checked native jobs in each orientation, including all actual +reference changes and all 436 relative helper endpoints. Cache reuse compares +the exact literal circuit and rank; a renumbered operator symbol alone is never +a cache key. + +For each actual helper entrance `E` and exit `U`, compute its literal relative +operator `H = T_U T_E^(-1)`. Its compiled normal form is + +``` +H = L C_w R, w = dim(U) - dim(E). +``` + +The shipped prewrapper is `R^(-1)` and the postwrapper is `L^(-1)`. Therefore + +``` +postwrapper * H * prewrapper = C_w tensor I_(9-w) +``` + +on every arbitrary dirty array, including the exact scalar phase. Coordinates +`0..w-1` are the normalized child coordinates; the other coordinates are literal +spectators. Every one of 436 wrappers is checked in both orientations. Another +224 actual monomial adapters per orientation identify the source `C_p` and +target `C_p^(-1) F` intrinsic data conventions. The global owner certificate +must place these actual normalized child coordinates into its pending block; +equal rank alone does not provide the owner chart. + +The finite native equality also specifies the repeated-coordinate lift: use the +`lambda`-fold tensor of each native factor. A scalar phase `i^a` in a finite +monomial then becomes `i^(a*lambda)`. Tensoring the checked equality preserves +the complete operator and its phase, with Fourier rank `w*lambda`; it does not +license reusing an unlifted scalar phase. The local free-operator proof is +universal in its operator inputs, so its scalar chronology is unchanged. + +## Exact fine-grid precision and active recursion depth + +The retained completed-child interface is exact Gaussian dyadic arithmetic; +there is no intermediate child rounding. A child on `r*f` axes has coefficients +in `2^(-r*f) Z[i]` and coordinate row-sum at most `2^(r*f)`. Inverse children use +the exact unit-phase identity. Only the outer completed layer truncates. The new +word preserves this interface: each scalar coefficient is dyadic, the actual +Horner realization uses only doubling, halving, addition and sign, and every +native monomial uses only permutations and unit Gaussian phases. + +Every compound ADD is explicitly expanded through owned coefficient work: +clear, build its integer numerator by Horner doubling/addition, halve for the +denominator, accumulate with sign, then clear. SCALE expands into doubling, +halving and negation; SWAP expands into three ADDs and negation. All these +operations and both final exact CLEAR postludes are charged in the fresh invoice. + +The noncancellation ideal envelope gives `P<=2^62` forward and `P<=2^63` +backward in stream Hilbert norm, including every private/work intermediate. +The 72 complete local calls and 72 bridge sweeps have normalized Hilbert norm +at most two each. For `lambda>=1`, conversion of an input coordinate bound to +Hilbert norm costs at most `37*lambda`, so the fresh magnitude guard is + +``` +63 + 144 + 37*lambda <= 244*lambda < 512*lambda. +``` + +The coefficient 512 is charged per lambda. Denominator exponents and recursive +temporary magnitudes are charged separately through the retained exact active- +depth argument. For the literal rank mass `s`, fresh nonchild charge `E`, +`B=s+E>=8`, and largest proper child rank `rmax<=72`, that argument is + +``` +A(e) <= A(rmax*floor(e/73)) + s*floor(e/73) + E, +A(e) <= 2*B*e, +A_layer(d) <= (2*B+18)*d < 32*73*B^2*d. +``` + +The fresh integer invoice checks `32*73*B^2 < 2^210000`, so the retained +common grid `Delta=ceil(2^210000*d)` covers this new word. Completed children, +rather than their internal temporaries, are composed; there is no multiplication +of each temporary bound by the number of sibling calls. This uses the retained +all-size exact compiler/child contract, not a strengthened error hypothesis. + +## Optional numerical robustness theorem + +The local precision receipts additionally prove a sufficient feedback bound +under their explicitly stated relative Hilbert-defect model. This is an +auxiliary implication; the exact fine-grid construction does not assume that +rounded-child model. Both real arithmetic component errors and all private +array errors are propagated through the expanded Horner/SWAP program. + +| Orientation | Ideal all-cut bound P | Defect propagation bound K | Sufficient local guard | +|---|---:|---:|---:| +| Forward | `P<=2^62` | `K<=2^65` |130 bits| +| Backward | `P<=2^63` | `K<=2^67` |133 bits| + +For initial maximum norm `B>=0`, linearity yields +`Emax<=K*u*(P*B+Emax)`. The recorded guard `p+k+3` implies +`Emax<=B*2^(-b)`, strictly for `B>0`; the relative model forces zero at `B=0`. +A separate coarse 506-per-lambda composition guard is recorded conditionally +on those defect contracts. Neither that optional model nor a coordinatewise +rounded-child implementation is asserted to be an inherited project theorem. + +## Fresh finite work and stock invoice + +The invoice binds the actual 31,392-row canonical owner table and all 72 complete +calls. It retains all 492 old local helper reserves, adds all 75 cleanup arrays, +and keeps eight centre and two coefficient-work reserves: + +``` +physical families =7168 +492 +75 +8 +2 =7745. +``` + +This deliberately charges more than the actual436 helper roles. Actual scalar +counts, including the exact private CLEAR postlude, are 162,954 real-component +primitives forward and 163,618 backward. Forty forward calls and thirty-two +backward calls total 11,753,936 such primitives. Routing/selection, every wrapper, +all literal instructions and the coefficient work are separately overcharged +through the retained `64*74^3` chart-compiler convention. The local primitive +bound has 44 bits and is strictly below `2^48`. + +Fresh integer evaluation gives: + +| Finite quantity | Fresh value | Retained bound | +|---|---:|---:| +| Physical stock bits |2641| source-counted | +| Logical work bits |2680|6050| +| Router work bits |7985|30000| +| Literal work bits |35282|100000| +| Physical row coefficient |16467|20161| +| Required C0 bits |200012|210000| + +These charges use the existing m=73 cover order and chart/row conventions with +the new literal program and new stock. No old invoice is copied as though it +checked the new word. The retained all-size compiler, physical route/tape, +exact completed-child common-grid and analytic-transfer hypotheses remain explicit. +The exact moment and all 47 outer inequalities are a separate assembly check. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/PROOF-OWNERS.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/PROOF-OWNERS.md new file mode 100644 index 0000000000..85f444e71c --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/PROOF-OWNERS.md @@ -0,0 +1,135 @@ +# Whole-family ownership and physical charts + +The owner certificate `owners/OWNERS.json.gz` contains all **31,392 helper roles** +in the 72 inherited invocations of the selected local word. Each invocation uses +436 complete helper families and retains the parent's 224 literal active data +families. The source-bound check is: + +```sh +python -I -O -B owners/verify.py +``` + +Run that command from this package directory. It reads only package inputs. An +optional `--out /path/outside/this/package` writes a fresh receipt and regenerated +chart certificate outside the package; an existing receipt is never overwritten. +Python 3.10 or later is required. The verifier uses only the standard library and +has no optimizer, solver, network, personal-library or Lean dependency. Explicit +checks remain enabled with Python's `-O` flag. + +This proves the finite ownership and chart assembly below, conditional on the +retained exact chart/cover semantics and the local operator identities checked +separately in `PROOF-LOCAL.md`. It does not establish the all-size realization, +recursive work, precision, tape or analytic transfer interfaces by itself. + +## Immutable inputs and actual time windows + +`owners/SOURCES.json` records the source hashes and public ancestry. The actual +hosted parent is `owners/inputs/HOSTED-PARENT.json.gz`, SHA-256 +`f713f8e66aaeaebb8202fe72bf36c7d23497152497342c27892e0fda3092f95b`. +The source DATA-PHASES is SHA-256 +`c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa`. +The selected forward trace is SHA-256 +`a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee`. +The canonical owner bytes are SHA-256 +`892b49f97d4caa7e92918262ea3a8cfa8cbf273c92d574ae53bb89a61ee5e847`. +These pins are checked before parsing the corresponding inputs. + +The verifier walks the complete serialized parent execution schedule. A local +invocation touches its data pair; each bridge touches both source and destination +pairs. For every inherited idle child it records the last physical touch and the +child's execution position. The supplied PLAN's allowed call indices must equal +the strictly intervening invocations, exactly. This reconstructs the windows from +the actual schedule, including every bridge, rather than trusting a PLAN interval. +The 160 derived windows include the zero-rank child entries; the PLAN records +only the 144 nonzero windows. + +Every helper must lie wholly inside its assigned window and on a family absent +from all 224 active data families of that invocation. Family identities are +literal, not interchangeable rank counters. The public ABI is + +```text +copy = owner_replica * 112 + port +family = owner_replica * 896 + (2 * pair + side) * 112 + port. +``` + +The checker reconstructs and verifies both fields. It also checks exact role +completeness and at most one helper per whole physical family in an invocation. + +## Actual chart construction + +For each port, the actual DATA-PHASES `q` is first required to equal the retained +local word's port label. To construct its orthogonal 73-column binary basis, +choose the first zero bit `z` of `q` among bits 0 through 8, put +`a = q xor (1 << z)`, and move index `z` to the front of the nine local indices. +For these ordered indices `i`, use the vectors +`(1 << i) xor a` when bit `i` of `a` is set, and `1 << i` otherwise. Append +`1 << i` for indices 9 through 72. The checker verifies all 73 squared norms and +all pairwise binary inner products. Every literal inherited pending-child axis +must belong to this basis, with the exact inherited rank and no repetitions. + +A width `d` helper's actual endpoint frames must be nested, and their dimension +difference must equal `d`. The separately checked whole-helper normalizer gives, +with its exact scalar phase, + +```text +post * (T_final * inverse(T_initial)) * pre = C_d tensor I_(9-d). +``` + +The chart sends the `d` normalized coordinates to the helper's exact assigned +contiguous block of the pending child's axis list. It chooses the first `9-d` +unused actual basis axes as explicit spectators, and appends all remaining axes. +The resulting 73 indices must be a permutation. The checker regenerates all +7,150 distinct physical charts and every complete helper route, and compares the +complete payload with `owners/CHARTS.json.gz`. + +The normalized whole-helper action therefore contributes precisely the assigned +pending coordinate factors and restores its spectators. Local helper inputs are +arbitrary inputs; the 76 separately owned private fields of the local trace do not +silently replace borrowed inputs by zero. Each invocation completes before the +next one begins. Intermediate spectator use consequently creates no concurrent +physical operation on the same family, even when different invocations use +different complete charts. + +## Exact inherited parent action + +Within each literal `(owner_replica, child, port)` the assigned axis blocks must +be pairwise disjoint and form an exact prefix. The residual child is the exact +suffix of the same inherited list. These commuting coordinate factors multiply +to the original child factor. Blocks are never transferred between different +children or families because their dimensions happen to match. + +Each helper executes after the preceding physical-family touch and before the +pending child. Moving its completed factor to that child's location crosses no +physical-family operation except other completed factors on disjoint orthogonal +coordinates, which commute. The whole-helper identity is the identity on every +spectator. Thus composing the completed helpers and the residual suffix preserves +the original parent's physical data action under the retained exact semantics. + +**All child phase and translation boundaries remain inherited, including children +whose residual rank is zero.** A rank-zero residual has no paid coordinate factor; +it does not authorize deleting its `translation_after` or other child metadata. +The complete unchanged hosted parent and DATA-PHASES carry these boundaries. +`owners/CHARTS.json.gz` is a coordinate-factor view, not a replacement serialized +parent program. The independent family-operator replay additionally checks the +complete factor partition and the zero-rank boundaries. + +## Recount and controls + +The selected local profile multiplied by 72 plus every actual residual suffix is + +```text +{1:176252, 2:30096, 3:30908, 4:10152, 5:4493, 6:18, + 7:494, 8:576, 32:11, 39:7, 48:2493}. +``` + +Its rank mass is 520704, its inherited live stock is 7168, and its deficit is +`73*7168 - 520704 = 2560`. All counts and the entire deterministic receipt are +recomputed and compared with `owners/EXPECTED.json`. The profile is an actual +combined invoice; it is not a sum of separately advertised gains. The exact +moment and outer arithmetic, and the global work/precision invoice, are checked +separately. + +Five negative controls alter the copy index, family, offset, width, or complete +role cardinality. Each must fail the same mathematical checker, without relying +on a mismatched input hash. Input-byte pins additionally reject changed retained +sources before certificate replay. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/PROOF.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/PROOF.md new file mode 100644 index 0000000000..aefa25b24f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/PROOF.md @@ -0,0 +1,117 @@ +# Sufficient assembly for kappa =3361/2000000 + +## Exact local and physical construction + +Use the44-selection in `local/inputs/WITNESS.json`. The complete local induction +and exact Gaussian native checks establish the two required endpoint shears, +restore arbitrary dirty helpers, and return all owned private fields to zero. +The local paid profile is + +``` +{1:2424,2:418,3:429,4:141,5:42,8:8}, mass5385. +``` + +The helper inventory uses actual relative endpoint width, including nonzero +entrance frames: `{4:40,5:4,7:27,8:125,9:240}`. The opposite orientation is the +complete reflected adjoint word, including COPY/CLEAR projectors and phases. +A naive adjoint substitution is not used. `PROOF-LOCAL.md` gives the universal +free-operator induction and the tensor lift, with scalar phase i^a lifted to +i^(a*lambda). Every helper includes actual pre/post monomial normalizers. + +The owner certificate places all 72*436 helper actions in actual pending physical +children of the retained T 8 parent, using disjoint literal coordinate blocks, +full untouched-family windows, and 73-column orthogonal charts. Invocations +serialize; each whole-helper operator restores all spectator coordinates. +Within the same physical family its normalized Gaussian axis factors commute. +The independent audit checks 387072 original factors, partitioned as 254088 hosted +and 132984 residual, with every retained translation boundary. It preserves +physical family tags, directions and exact phases. Thus the completed helpers +and residual children reproduce the original parent action. A global bag of +axis ranks or untagged factors would not suffice; this avoids the withdrawn +cross-family axis-overlay error of PR1324/1305. + +## Finite invoice and retained exact contract + +The construction retains 7168 data families and all 492 old helper reserves, +then charges 75 cleanup arrays,8 centres and 2 coefficient-work reserves, giving +7745 families. Actual Horner, SWAP and dyadic primitive work is recounted. +The exact completed-child interface and active-depth recurrence yield the +common fine-grid coefficient bound, with fresh C 0<2^210000. The magnitude guard +is 244*lambda<512*lambda, and the new row coefficient 16467<20161. The finite +invoice reuses the existing chart/cover conventions but evaluates the new +literal word and stock. It does not multiply internal child temporaries across +siblings or truncate at child boundaries. + +For B=s+E>=8 and largest proper child rank rmax<73, +A(e)<=A(rmax*floor(e/73))+s*floor(e/73)+E gives A(e)<=2Be by induction. +Indeed, for f=floor(e/73)>=1, 2B(73-rmax)>=s+E, and the base leaf has A(e)<=8e. +The retained disjoint outer decomposition plus 18 d extra units gives +A_layer(d)<32*73*B^2*d. The fresh invoice verifies the chosen C 0 dominates this. +The retained all-size compiler must implement the lifted local operations and +physical charts under those existing exact interfaces. This is a sufficient +conditional assembly, not a new proof of that entire compiler. + +## Exact recurrence saving + +Fresh recount from the raw owners and local trace gives + +``` +{1:176252,2:30096,3:30908,4:10152,5:4493,6:18, + 7:494,8:576,32:11,39:7,48:2493}. +``` + +Here m=73, W=7168, rank mass 520704 and deficit 2560. The retained halving guard +73^2>2*48^2 holds. The Mercator engine with 224-bit dyadic enclosures and the +independent 80-term atanh engine both enclose the complete paid moment strictly +below1 at b=8417/5000000. Both the native moment and the more conservative +retained bad-residue surcharge pass. These engines import no producer profile; +the outer checker independently recounts the full profile from literal owners. +The unchanged bit rail is checked at +c=1733427039589844234749/10^24, including its old surcharge and exact deficit. +Its circuit/realization assumptions remain inherited. + +## Fixed-depth bootstrap and finite constants + +Take eta=beta=10^-8 and s=min(c,(1-beta)*b)-10^-11. +Let a0=384599/10^10 and aj=(1-s)*s+s*a(j-1) for j=1,...,7. +The retained stopped construction, calling only the preceding completed level, +has normalized cost + +Tj(e)/V=O(e^(1-s)+e^((1-s)^2+s*(1-a(j-1)))+e^s log(e)). + +The seven exact positive stage gaps checked by the script imply ordinary saving +aj. Coarse children decrease width; ordinary calls decrease j; a 0 never calls +a new level. Seven is a fixed finite depth, so fixed tape counts compose. + +For each stage choose a finite constant C dominating its fixed implementation, +primitive, setup, wrapper, row and previous completed-level constants. This +exists under the inherited completed-interface hypotheses. Choose an integer +ell with C<2^ell, and let g be the minimum positive stage gap. The sufficient +threshold recipe is + +L=max(ceil(36/g^2),ceil(2*(4+ell)/g)), e0=2^L. + +For t=log2(e)>=L, 2log2(t)<=3sqrt(t)<=g*t/2 and4+ell<=g*t/2, +so C*16*t^2<=2^(g*t). Absorb the fixed number of lower-order terms into C; +use the preceding algorithm below e 0 and choose the next finite constant +sequentially. This is not circular and is independent of the input size. +The checker records example thresholds for ell 512 only as arithmetic controls; +512 does not bound the actual implementation constants or C 0. An actual +threshold must use its own dominating ell at each stage. + +## Outer assembly and conclusion + +Feed a 7,b to the unchanged balanced-prefix outer assembly with C 1=1, +C0=2^210000, literal gap1 and degree gap10^6-(51/25)*20161>0. +The exact checker recomputes all 47 strict constraints and all seven performance +margins. Each exceeds kappa=3361/2000000 where required. The controls reject +zero eta, zero beta, beta 1, a kappa above the balanced margin, and a supplier +saving whose paid moment already exceeds 1. No decimal optimization is proof +input; all admission arithmetic uses exact rational intervals. + +Under the retained all-size exact complex/bit interfaces, physical fixed-tape +realization, completed ordinary bootstrap, exact recovery and analytic transfer, +this construction therefore supplies the project bound with kappa=0.0016805. +These inherited premises and their publication status are unchanged. The claim +is the new checked conditional construction and exponent, not an unconditional +integer multiplication theorem or a full proof-assistant formalization. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/README.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/README.md new file mode 100644 index 0000000000..48809a8500 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/README.md @@ -0,0 +1,83 @@ +# Selective 44 borrowing: conditional kappa =0.0016805 + +This package constructs and checks a new finite complex supplier and derives +`kappa=3361/2000000=0.0016805` under the project's retained all-size and analytic +interfaces. This exceeds the highest competing open conditional claim found in +the recorded review, approximately 0.0016797549561255445 (PR 2220), by about +0.000000745043874455, or 0.0443543%. The maintainer-selected bound is separately +0.001672660197076918. No selected pointer is changed. + +The construction selects 44 high dirty-reference aliases, retaining the four +high helpers in block 632, and completes both native orientations of PR 2226's +anticipatory borrowing architecture. It has 436 physical helper roles per call, +75 new cleanup buffers, and a local paid rank mass 5385. New multi-bin repacking +hosts all 31,392 roles of the actual 72-call T 8 controller. The independently +recounted complex saving is at least 0.0016834, even with the retained conservative +bad-residue surcharge. This gain changes the paid construction rather than +refining digits of a predecessor's moment root. + +## Reproduce + +Python 3.10+ and the standard library suffice for certificate replay. From the +repository root: + +```sh +python3 -I -O -B research/selective-borrowing-kappa/verify.py +``` + +For persistent receipts, append `--out /an/unused/directory/outside/the/package`. +The source package remains byte-identical. The registered Python 3.12 hook is: + +```sh +python3.12 scripts/research_package.py --package research/selective-borrowing-kappa +``` + +`VERIFY.json` registers this exact command for CI. The four-agent local campaign +used CPython 3.11.9; a Python 3.12 or hosted CI pass is not asserted in advance. +Optional `local/rebuild.py` regenerates the44-selection traces; certificate replay +needs neither SciPy nor a solver. Finite replay usually takes under two minutes. + +## Proof and audit map + +- `PROOF.md`: sufficient composition and exact bootstrap/outer assembly. +- `PROOF-LOCAL.md`: both complete native words, arbitrary dirty inputs, actual + helper and data adapters, tensor phases, exact common-grid precision and stock. +- `PROOF-OWNERS.md`: literal physical families, chronology, charts and residuals. +- `independent/PROOF.md`: separate family-tagged Gaussian factor partition. +- `arithmetic/verify.py`: raw owner recount, two independent rational moment + engines, retained bit rail, seven-level bootstrap, all 47 strict inequalities. +- `lean/Factors.lean` and `lean/Feedback.lean`: eleven Mathlib-only universal + lemmas, independently kernel/axiom checked. These establish algebraic + reordering and an optional feedback implication; they do not formalize the + entire finite certificate or the integer multiplication theorem. + +All 5992 actual native jobs,436 helper normalizers plus 224 data adapters in each +orientation, all 660 public input columns, both 76-CLEAR owned-private postludes, +7150 literal 73-column charts, and 387072 parent factors are checked. Local, owner, +physical-factor and outer fault controls reject. `MANIFEST.json` binds every +published file; source pins bind the public predecessors. + +## Scope + +This is a conditional finite supplier contribution. It supplies the changed +local/native/hosting/work/stock/precision evidence and exact arithmetic assembly. +The project's all-size common-label and mutable-source realization, physical +routing and fixed-tape compiler, retained bit construction, exact recovery and +analytic transfer remain inherited premises. Exact Gaussian dyadic arithmetic +is used inside children, with the outer truncation boundary retained. The +auxiliary relative-error envelope introduces no rounded-child assumption into +admission. No unconditional multiplication theorem, complete Lean proof, +independent human review or global optimality is claimed. + +The broad recent review and final checkpoint are recorded in `FRONTIER.json`. +It covers PR metadata and targeted relevant source review, not every discussion +or arbitrary diff. Upstream credit and licenses are in `NOTICE.md` and retained +notices. No personal Lean tree or original machine path is imported. + +## Open next questions + +Can another alias selection or joint bin dissolution improve the fully hosted +profile? Change one input in a separate experimental copy and rerun the entire +admission. Can the finite Python certificates be reflected into Lean? The +included universal lemmas provide an algebraic starting point, without claiming +that reflection has already been implemented. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/SOURCES.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/SOURCES.json new file mode 100644 index 0000000000..b4d2a8a572 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/SOURCES.json @@ -0,0 +1,24 @@ +{ + "schema_version": 1, + "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11", + "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "e8870e5917057d59b730eb14721a778639eca050", + "sha256": "d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7" + } + ], + "retained_sources": { + "local": "local/SOURCE-PINS.json", + "owners": "owners/SOURCES.json", + "physical_factors": "independent/SOURCES.json", + "arithmetic": "arithmetic/SOURCES.json" + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/TASK.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/TASK.json new file mode 100644 index 0000000000..f705c894a7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/TASK.json @@ -0,0 +1,64 @@ +{ + "schema_version": 1, + "task_id": "complex-new-helper-inventory", + "priority_revision": 4, + "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2", + "original_card": { + "id": "complex-new-helper-inventory", + "question": "Can a new local cost interface change the deficit-to-recursive-cost ratio beyond current packing?", + "starting_points": [ + "research/complex-local-word-ceiling/README.md", + "research/heterogeneous-owner-assembly/README.md", + "certificates/selected-result.json", + "research/selected-k8-unrestricted-ends/README.md" + ], + "methods": [ + "Propose a changed helper inventory and derive an optimistic exact moment.", + "Design a small completed instance and independently verify its endpoint and rank bill." + ], + "acceptance_gate": "A new physically realizable contract outside the frozen catalogue whose per-call profile, screened with ceiling.ceiling(), beats the selected word's packing ceiling; a lower cap does not establish identical circuitry or order attainable optima, and is insufficient to pass the stronger-budget target under this model.", + "first_step": { + "command": [ + "python3", + "-B", + "research/complex-local-word-ceiling/verify.py" + ], + "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING", + "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.", + "scope": "Positive control and planning budget only; this is not a new construction or full cold replay." + }, + "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin.", + "bottleneck_link": "Packing of the selected word is capped at +0.87% (ceiling model); only a lower local excess spend or a changed coverage law raises the cap.", + "allowed_variations": [ + "Narrow to one changed boundary or helper type with explicit counts.", + "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope." + ], + "scope_change_triggers": [ + "Mixing only providers already inside the pinned catalogue.", + "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013." + ], + "overlap_terms": [ + "helper inventory", + "heterogeneous-owner", + "mutable leaver" + ] + }, + "scope_status": "changed", + "actual_scope": "Complete exact finite selective44 borrowing supplier, physical T8 owner charts and conditional kappa=0.0016805 assembly.", + "scope_change_reason": "User-directed four-agent campaign extends the public dirty-reference architecture with a selective44 native word and actual global ownership and paid moment; no random task draw is claimed.", + "bottleneck_link": "Changes paid local profile and residual distribution rather than refining digits of the prior supplier.", + "budget_minutes": 60, + "stop_rule": "Freeze only after native both-direction replay, physical factor partition, finite exact precision/work invoice, two independent rational moments and all47 strict outer inequalities pass.", + "outcome": "completed", + "evidence": "verify.py, PROOF.md, PROOF-LOCAL.md, PROOF-OWNERS.md and independent/PROOF.md", + "author": "Dugongue", + "ai_assistance": "Substantial OpenAI Codex assistance by one primary agent and three proof/review agents. No independent human review or full Lean multiplication formalization is claimed.", + "open_questions": [ + { + "question": "Can the selective44 packing or local word be improved further?", + "why_it_matters": "The new verified physical profile raises the available complex saving.", + "first_step": "Change local/inputs/WITNESS.json or the owner table in a separate experiment, then replay the complete native and physical certificate and exact moment.", + "blocked_by": "No optimality claim is made for the bounded search used to find this witness." + } + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/VERIFY.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/VERIFY.json new file mode 100644 index 0000000000..565eb9076f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/VERIFY.json @@ -0,0 +1,72 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/selective-borrowing-kappa/FRONTIER.json", + "research/selective-borrowing-kappa/SOURCES.json", + "research/selective-borrowing-kappa/TASK.json", + "research/selective-borrowing-kappa/arithmetic/EXPECTED.json", + "research/selective-borrowing-kappa/arithmetic/SOURCES.json", + "research/selective-borrowing-kappa/arithmetic/vendor/BIT-EXPECTED.json", + "research/selective-borrowing-kappa/arithmetic/vendor/base_two_moment.py", + "research/selective-borrowing-kappa/arithmetic/vendor/moment.py", + "research/selective-borrowing-kappa/arithmetic/vendor/outer.py", + "research/selective-borrowing-kappa/arithmetic/verify.py", + "research/selective-borrowing-kappa/independent/EXPECTED.json", + "research/selective-borrowing-kappa/independent/SOURCES.json", + "research/selective-borrowing-kappa/independent/audit.py", + "research/selective-borrowing-kappa/independent/inputs/T8-PARENT.json.gz", + "research/selective-borrowing-kappa/independent/verify.py", + "research/selective-borrowing-kappa/lean/CHECKS.json", + "research/selective-borrowing-kappa/lean/Factors.lean", + "research/selective-borrowing-kappa/lean/Feedback.lean", + "research/selective-borrowing-kappa/local/SOURCE-PINS.json", + "research/selective-borrowing-kappa/local/check_trace.py", + "research/selective-borrowing-kappa/local/data/backward.json", + "research/selective-borrowing-kappa/local/data/expected-local.json", + "research/selective-borrowing-kappa/local/data/finite-invoice.json", + "research/selective-borrowing-kappa/local/data/forward.json", + "research/selective-borrowing-kappa/local/data/native-backward.json.gz", + "research/selective-borrowing-kappa/local/data/native-forward.json.gz", + "research/selective-borrowing-kappa/local/data/postludes.json", + "research/selective-borrowing-kappa/local/data/precision-backward.json", + "research/selective-borrowing-kappa/local/data/precision-forward.json", + "research/selective-borrowing-kappa/local/data/wrappers-backward.json", + "research/selective-borrowing-kappa/local/data/wrappers-forward.json", + "research/selective-borrowing-kappa/local/inputs/WITNESS.json", + "research/selective-borrowing-kappa/local/inputs/selected-block-certificate.json", + "research/selective-borrowing-kappa/local/inputs/selected-complex-word.json.gz", + "research/selective-borrowing-kappa/local/invoice.py", + "research/selective-borrowing-kappa/local/native.py", + "research/selective-borrowing-kappa/local/precision.py", + "research/selective-borrowing-kappa/local/rebuild.py", + "research/selective-borrowing-kappa/local/vendor/PR2226-replay.py", + "research/selective-borrowing-kappa/local/vendor/clifford_exact.py", + "research/selective-borrowing-kappa/local/vendor/native_verify.py", + "research/selective-borrowing-kappa/local/verify.py", + "research/selective-borrowing-kappa/owners/CHARTS.json.gz", + "research/selective-borrowing-kappa/owners/EXPECTED.json", + "research/selective-borrowing-kappa/owners/OWNERS.json.gz", + "research/selective-borrowing-kappa/owners/SOURCES.json", + "research/selective-borrowing-kappa/owners/inputs/DATA-PHASES.json", + "research/selective-borrowing-kappa/owners/inputs/HOSTED-PARENT.json.gz", + "research/selective-borrowing-kappa/owners/inputs/PLAN.json", + "research/selective-borrowing-kappa/owners/verify.py", + "research/selective-borrowing-kappa/verify.py" + ], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-O", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 600, + "checks": "Complete native forward/backward,660 public columns and76 owned-zero fields;5992 exact native jobs;436 helper+224data adapters each orientation;31392 whole owners/7150 charts;387072 physical tagged factors; exact fine-grid work/stock invoice; two independent rational moment engines; all47 strict outer rows and mutation controls. 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"11328020179687499753837821390398475304756545994729490696711051065815214613516192581347024338307300851247194226959/3906250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "Gaussian_arithmetic_above_kappa": "2335168075766401646696581767784460605647139393026368251177639034694537463297820839808970605844261663908613134758172373548962879/31302606248109312516769280702610509136379974296319196788566416969897052753546377695202488185381690934512680809417616120923686560000000", + "scalar_work_above_kappa": "12458405535034741046402140856487565756645196322539821175602355597720113635327763641424915233855850592905882694026302270767906559/156513031240546562583846403513052545681899871481595983942832084849485263767731888476012440926908454672563404047088080604618432800000000", + "dimension_above_kappa": "389967445721250653753944615297240787992453918164937944872460176703526100035595816403540152732555825854806892998747171201359846809199/391282578101366406459616008782631364204749678703989959857080212123713159419329721190031102317271136681408510117720201511546082000000" + }, + "controls_rejected": [ + "zero_eta", + "zero_beta", + "unit_beta", + "above_balance", + "supplier_endpoint_too_large" + ], + "source_sha256": { + "owners/OWNERS.json.gz": "892b49f97d4caa7e92918262ea3a8cfa8cbf273c92d574ae53bb89a61ee5e847", + "owners/EXPECTED.json": "c4ad5debab3f3bd5cba68b91e28a6a8fa64febcbeb892b1fd1d27e3977a76cb1", + "local/data/forward.json": "a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee", + "owners/inputs/PLAN.json": "7b3122026e82b9d9e964d2a50c7bffdc976ae3dfea64bb98b2fdad5cfbdbaa20", + "arithmetic/vendor/moment.py": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482", + "arithmetic/vendor/base_two_moment.py": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098", + "arithmetic/vendor/outer.py": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c", + "arithmetic/vendor/BIT-EXPECTED.json": "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4" + }, + "scope": "Exact finite recount and retained conditional outer arithmetic only. Requires independently source-bound native word, physical owner charts/controller, full work/precision/stock invoice and original all-size/analytic/bit assumptions; not full Lean multiplication proof." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/SOURCES.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/SOURCES.json new file mode 100644 index 0000000000..ec41c61c7b --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/SOURCES.json @@ -0,0 +1,25 @@ +{ + "upstream_revision": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2", + "files": [ + { + "path": "vendor/moment.py", + "sha256": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482", + "source": "https://github.com/CrocSwap/integer-mult-bounds/blob/0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2/research/frame-domain-ceiling/engines/moment.py" + }, + { + "path": "vendor/base_two_moment.py", + "sha256": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098", + "source": "https://github.com/CrocSwap/integer-mult-bounds/blob/0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2/research/frame-domain-ceiling/engines/base_two_moment.py" + }, + { + "path": "vendor/outer.py", + "sha256": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c", + "source": "https://github.com/CrocSwap/integer-mult-bounds/blob/0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2/research/complex-multicold/engines/outer.py" + }, + { + "path": "vendor/BIT-EXPECTED.json", + "sha256": "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4", + "source": "https://github.com/CrocSwap/integer-mult-bounds/blob/0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2/research/e8-bit-p111-tuned/tune/EXPECTED.json" + } + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/vendor/BIT-EXPECTED.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/vendor/BIT-EXPECTED.json new file mode 100644 index 0000000000..e293ddc557 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/vendor/BIT-EXPECTED.json @@ -0,0 +1,222 @@ +{ + "word": { + "file": "tune/bitE8mut-p111-tuned.json.gz", + "file_sha256": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f", + "content_sha256": "5642fd0e35a1acc11acff8d0c08350c1617e95f310d4d35ad63a651f9eeeba0b", + "v": 111, + "R": 439, + "punctured": [ + 10, + 15, + 51, + 53, + 56, + 66, + 72, + 89, + 111 + ], + "retained_totals": 9, + "copy_ranks": [ + 7, + 7, + 7, + 7, + 7, + 7, + 7, + 7, + 7 + ], + "decoder_constant": 63, + "gates": { + "D": 111, + "A": 2075, + "B": 712, + "C": 2574 + }, + "local_histogram": { + "1": 1323, + "2": 1342, + "3": 79, + "4": 44, + "5": 102, + "6": 11, + "7": 19 + }, + "legality": { + "records": 27234, + "common_frame_ADDs": 24305, + "rank_mass": 5129 + }, + "five_stage_columns": 883, + "five_stage_payload_bits": 93, + "nine_stage_columns": 1327, + "nine_stage_payload_bits": 93, + "eleven_stage_columns": 1549, + "eleven_stage_payload_bits": 93, + "base_content_sha256": "d0df3c89ae84989f4045ba1681a5e2e9fcdfa38ed0301c8abf5185537155cbe3" + }, + "interface": { + "hypotheses": { + "H1_no_target_feeds_z": true, + "H2_D_frame0_targets_from_helpers": true, + "H3_A_z_only_scatter_targets_B_split": true, + "H4_C_literal_reverse_at_full": true, + "H5_V_identity": true + }, + "conclusion": { + "U_block_form": true, + "J_identity": true, + "K_equals_D": true, + "C_is_U_inverse": true + }, + "interleaving": { + "z_gates_A": 2075, + "source_writes_A": 0, + "z_gates_B": 499, + "source_writes_B": 483, + "helper_writes_B": 16, + "sources_rewritten_B": 108, + "B_z_write_frame_dims": { + "4": 388, + "6": 111 + }, + "B_z_operand_kinds": { + "helper<-helper": 3, + "helper<-source": 13, + "source<-helper": 108, + "source<-source": 375 + }, + "B_z_operand_kinds_by_dim": { + "dim4:helper<-source": 13, + "dim4:source<-source": 375, + "dim6:helper<-helper": 3, + "dim6:source<-helper": 108 + }, + "B_z_writes_after_a_target_read_of_the_same_register": 454, + "B_target_reads_of_registers_rewritten_earlier_in_B": 111, + "target_reads_of_targets": 0, + "cleanup_gates": 2574 + } + }, + "nine": { + "pipeline": "PR #510 (PR #463 nine-stage template, 32 replicas, PR #442 hosting, PR #493 outer assembly), parameterised for v <= 120 by code/adapt9p.py", + "bit_lower": "1733427039589844234749/1000000000000000000000000", + "bit_lower_decimal": "0.001733427039589844234749", + "profile": { + "1": 381024, + "2": 400416, + "3": 22752, + "4": 26880, + "5": 29376, + "6": 17376, + "7": 26784, + "42": 288 + }, + "bit": { + "live": 28416, + "physical": 28425, + "maxchild": 42, + "rank_mass": 1808352, + "hosted_residuals": 126432, + "bank_assignments": 0, + "prefix_bits": 177, + "charged_prefix_bits": 192, + "coefficient_bits": 177, + "two_level_coefficient_bits": 393, + "charged_two_level_coefficient_bits": 512, + "row_coefficient": 18355, + "halving_depth": 2 + }, + "kappa": { + "pr548": { + "kappa": "49718254847979/31250000000000000", + "kappa_decimal": "0.001590984155135328", + "complex_lower": "796759709641153381751/500000000000000000000000", + "binding_component": "complex" + }, + "pr544": { + "kappa": "397677800014309/250000000000000000", + "kappa_decimal": "0.001590711200057236", + "complex_lower": "31864911873350496687/20000000000000000000000", + "binding_component": "complex" + }, + "pr535": { + "kappa": "793733736881443/500000000000000000", + "kappa_decimal": "0.001587467473762886", + "complex_lower": "1589991533606044174031/1000000000000000000000000", + "binding_component": "complex" + }, + "pr531": { + "kappa": "1517760977820331/1000000000000000000", + "kappa_decimal": "0.001517760977820331", + "complex_lower": "304013615566499291631/200000000000000000000000", + "binding_component": "complex" + } + } + }, + "eleven": { + "pipeline": "PR #493 (PR #445 geometry, PR #472 emitter, audit and invoice, PR #442 hosting), parameterised by PR #545's code/adapt.py", + "bit_lower": "855475950388791987811/500000000000000000000000", + "bit_lower_decimal": "0.001710951900777583975622", + "profile": { + "1": 116424, + "2": 121648, + "3": 6952, + "4": 7424, + "5": 8976, + "6": 4520, + "7": 8776, + "56": 2600 + }, + "bit": { + "live": 8880, + "physical": 8889, + "maxchild": 56, + "rank_mass": 689304, + "hosted_residuals": 38632, + "prefix_bits": 4304, + "charged_prefix_bits": 4352, + "coefficient_bits": 4304, + "three_level_coefficient_bits": 12966, + "charged_three_level_coefficient_bits": 12992, + "row_coefficient": 19431, + "halving_depth": 3 + }, + "emitted_program_sha256": "793b6f40c9aa4b8db57fec0ac0973e16884dae28707e482ee8f7a2349a5d5a4b", + "kappa": { + "pr548": { + "kappa": "49718254847979/31250000000000000", + "kappa_decimal": "0.001590984155135328", + "complex_lower": "796759709641153381751/500000000000000000000000", + "binding_component": "complex" + }, + "pr544": { + "kappa": "397677800014309/250000000000000000", + "kappa_decimal": "0.001590711200057236", + "complex_lower": "31864911873350496687/20000000000000000000000", + "binding_component": "complex" + }, + "pr535": { + "kappa": "793733736881443/500000000000000000", + "kappa_decimal": "0.001587467473762886", + "complex_lower": "1589991533606044174031/1000000000000000000000000", + "binding_component": "complex" + }, + "pr531": { + "kappa": "1517760977820331/1000000000000000000", + "kappa_decimal": "0.001517760977820331", + "complex_lower": "304013615566499291631/200000000000000000000000", + "binding_component": "complex" + } + } + }, + "base": { + "nine_bit_lower": "1733052823330162099157/1000000000000000000000000", + "nine_bit_lower_decimal": "0.001733052823330162099157", + "eleven_bit_lower": "427652201524503168379/250000000000000000000000", + "eleven_bit_lower_decimal": "0.001710608806098012673516", + "note": "PR #555's EXPECTED.json (untuned word); the tune/ pipeline is the same, with the same frozen hosting selections" + } +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/vendor/base_two_moment.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/vendor/base_two_moment.py new file mode 100644 index 0000000000..14a90fa0e7 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/vendor/base_two_moment.py @@ -0,0 +1,63 @@ +#!/usr/bin/env python3 +"""Maintainer arithmetic cross-check of the pinned PR39 certificate. + +Uses no producer/checker imports. Counts are inputs, not proved by this script. +80-term rational logarithms and 12-term exponentials independently enclose +the two moments. The accompanying written review addresses their realization. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[1] +GRID = 10**30 + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def log_unit(x): + require(1 <= x <= 2, "logarithm range") + z = (x-1)/(x+1) + lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0)) + return lo, lo+2*z**161/(161*(1-z*z)) + + +def log_bounds(x): + k = 0 + while x > 2: + x /= 2 + k += 1 + lo, hi = log_unit(x) + l2, u2 = log_unit(Q(2)) + lo, hi = lo+k*l2, hi+k*u2 + # Outward rational rounding keeps later certificates compact. + return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID) + + +def exp_bounds(x): + require(0 <= x < 1, "exponential range") + term = total = Q(1) + for j in range(1, 13): + term *= x/j + total += term + tail = term*x/13/(1-x/14) + return total, total+tail + + +def moment(m, W, entries, saving): + lo = hi = Q(0) + for t, count in entries: + t, count = int(t), int(count) + require(0 < t < m and count > 0, "improper child") + l, u = log_bounds(Q(m, t)) + lower = exp_bounds(saving*l)[0] + upper = exp_bounds(saving*u)[1] + lo += Q(t*count, m*W)*lower + hi += Q(t*count, m*W)*upper + return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID) + diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/vendor/moment.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/vendor/moment.py new file mode 100644 index 0000000000..7d697cb452 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/vendor/moment.py @@ -0,0 +1,113 @@ +"""Independent prospective ordinary five-stage cost, never supplier admission. + +Consumes a raw one-invocation ledger made from the actual source/internal/target +paths. No packed profile is used as an input. Missing physical/finite/compiler +obligations must remain explicit in the accompanying report. +""" +from collections import Counter +from decimal import Decimal, localcontext, ROUND_FLOOR +from fractions import Fraction as F +from functools import lru_cache +from math import factorial +import hashlib, json, sys +from pathlib import Path + +DYAD = 1 << 224 + +def floor(x): return F(x.numerator*DYAD//x.denominator, DYAD) +def ceil(x): return -floor(-x) + +@lru_cache(None) +def logarithm(x): + """Mercator bounds after binary reduction; ln(2) via two Mercator sums. + + For 1 <= y < 2, log(y) is an alternating series in z=y-1. + We instead reduce to 1 <= y < 3/2, and bound log(3/2) and + log(2) by the same alternating series at z=1/2 and z=1/3, + using log(2)=log(3/2)+log(4/3). All sums are exact rationals. + """ + assert x >= 1 + k = 0 + while x >= 2: x /= 2; k += 1 + j = int(x >= F(3, 2)) + if j: x /= F(3, 2) + def mercator(y): + z=y-1 + assert 0 <= z <= F(1,2) + # An even partial sum is below log; the next term bounds its tail. + t=z; s=F(0) + for n in range(1, 225): + s += (t if n & 1 else -t)/n + t *= z + return s, s+t/225 + a,b=mercator(x); c,d=mercator(F(3,2)); e,f=mercator(F(4,3)) + return floor(a+(k+j)*c+k*e), ceil(b+(k+j)*d+k*f) + +def exponential(l,u): + assert 0 <= l <= u < F(1,10) + def poly(x): + s=t=F(1) + for j in range(1,25): t=t*x/j; s+=t + return s,t + lo,_=poly(l);hi,last=poly(u) + return floor(lo),ceil(hi+last*u/25/(1-u/26)) + +def moment(H,m,W,a,bad=False): + lo=hi=F(0) + for r,n in H.items(): + assert 0 < r < m and n > 0 + l,u=logarithm(F(m,r)); e,f=exponential(a*l,a*u) + p=F(r*n,m*W);lo+=p*e;hi+=p*f + if bad: + # Existing envelope shape, only conditionally portable to the new cover. + l,u=logarithm(F(m));e,f=exponential(a*l,a*u) + p=F(32*m*sum(H.values()),10**16*W) + lo+=p*e;hi+=p*f + return floor(lo),ceil(hi) + +def decimal(x): + with localcontext() as c: + c.prec=70 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def certify(H,m,W,bad=False): + with localcontext() as c: + c.prec=70 + terms=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln()) for r,n in H.items()] + fallback=Decimal(32*m*sum(H.values()))/Decimal(10**16*W) if bad else Decimal(0) + lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003') + for _ in range(210): + a=(lo+hi)/2 + if sum(p*(a*l).exp() for p,l in terms)+fallback*(a*lm).exp()<1:lo=a + else:hi=a + q=int(((lo+hi)/2*10**24).to_integral_value(rounding=ROUND_FLOOR)) + lo=F(q,10**24);hi=lo+F(1,10**24) + lv=moment(H,m,W,lo,bad);hv=moment(H,m,W,hi,bad) + assert lv[1]<10 for v in slacks.values()),{k:str(v) for k,v in slacks.items() if v<=0} + assert min(margins.values())==minimum + assert margins['balanced_prefix']-minimum==eta and 1-eps-r==eta/2 + assert 1-eps*(1+c)==eta-eps*eta/4 + return dict(parameters=dict(a_bit=a,a_complex=b,tau=tau,sigma=sigma, + eta=eta,beta=beta,q=q,c=c,epsilon=eps,lambda_=lam, + lambda_prime=lp,alpha_squared_power=r,delta=delta, + C0=bridge['semantic']['C0'],C1=1,kappa=kappa), + strict_constraints=slacks,margins=margins,minimum_margin=minimum, + absorption_gap=minimum-kappa, + recurrence=dict(internal=internal,leaf=leaf,reservations=1-c)) diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/verify.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/verify.py new file mode 100644 index 0000000000..93448459db --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/arithmetic/verify.py @@ -0,0 +1,170 @@ +"""Fresh owner recount, two exact moments and unchanged 47-row outer assembly.""" +from pathlib import Path +from fractions import Fraction as Q +from functools import lru_cache +from collections import Counter, defaultdict +from decimal import Decimal, localcontext +import json, gzip, hashlib, importlib.util, argparse +import sys +sys.dont_write_bytecode = True +root = Path(__file__).resolve().parent +campaign = root.parent +parser = argparse.ArgumentParser(description=__doc__) +parser.add_argument('--out', type=Path, help='Optional external receipt file') +args = parser.parse_args() +if args.out and args.out.resolve().is_relative_to(campaign.resolve()): + raise ValueError('--out must be outside the published package') +for pin in json.loads((root / 'SOURCES.json').read_bytes())['files']: + if hashlib.sha256((root / pin['path']).read_bytes()).hexdigest() != pin['sha256']: + raise ValueError('arithmetic vendor source drift') +base = campaign / 'owners/inputs' +plan = json.loads((base / 'PLAN.json').read_bytes()) +owners_path = campaign / 'owners/OWNERS.json.gz' +audit_path = campaign / 'owners/EXPECTED.json' +trace_path = campaign / 'local/data/forward.json' +trace = json.loads(trace_path.read_bytes()) +rows = json.loads(gzip.decompress(owners_path.read_bytes())) +bycall = defaultdict(list) +bybin = defaultdict(list) +widths = {int(k): v for k, v in trace['helper_widths'].items()} +for r in rows: + if not r['copy'] == 112 * r['owner_replica'] + r['port']: + raise ValueError('exact arithmetic admission failed') + if not r['width'] == widths[r['helper']]: + raise ValueError('exact arithmetic admission failed') + if not plan['order'][r['time']] == [r['stage'], r['replica']]: + raise ValueError('exact arithmetic admission failed') + if not r['time'] in plan['windows'][f"{r['owner_replica']},{r['child']}"][1]: + raise ValueError('exact arithmetic admission failed') + bycall[r['time']].append(r) + bybin[r['owner_replica'], r['child'], r['port']].append(r) +if not (len(bycall) == 72 and len(rows) == 72 * 436): + raise ValueError('exact arithmetic admission failed') +for rr in bycall.values(): + if not {r['helper'] for r in rr} == set(widths): + raise ValueError('exact arithmetic admission failed') + if not len({r['family'] for r in rr}) == len(rr): + raise ValueError('exact arithmetic admission failed') +occupied = Counter() +for k, rr in bybin.items(): + cap = plan['windows'][f'{k[0]},{k[1]}'][0] + used = set() + for r in rr: + axes = set(range(r['offset'], r['offset'] + r['width'])) + if not (axes and min(axes) >= 0 and (max(axes) < cap) and (not used & axes)): + raise ValueError('exact arithmetic admission failed') + used |= axes + occupied[k] = len(used) +H = Counter() +for wc, (cap, times) in plan['windows'].items(): + rep, ch = map(int, wc.split(',')) + for port in range(112): + rem = cap - occupied[rep, ch, port] + if not rem >= 0: + raise ValueError('exact arithmetic admission failed') + if rem: + H[rem] += 1 +for r, n in trace['local_profile'].items(): + H[int(r)] += 72 * n +W = plan['live'] +m = 73 +mass = sum((r * n for r, n in H.items())) +if not (mass == 520704 and m * W - mass == 2560): + raise ValueError('exact arithmetic admission failed') +if not m * m > 2 * max(H) ** 2: + raise ValueError('exact arithmetic admission failed') + +def load(name, path): + sp = importlib.util.spec_from_file_location(name, path) + mod = importlib.util.module_from_spec(sp) + sp.loader.exec_module(mod) + return mod +eng = root / 'vendor' +M = load('exact_mercator', eng / 'moment.py') +B = load('independent_atanh', eng / 'base_two_moment.py') +B.log_bounds = lru_cache(None)(B.log_bounds) +outer_path = eng / 'outer.py' +A = load('retained_outer', outer_path) +_retained_assembly = A.assembly +def checked_assembly(*positional, **keywords): + result = _retained_assembly(*positional, **keywords) + if len(result['strict_constraints']) != 47 or len(result['margins']) != 7: + raise ValueError('retained strict row inventory changed') + if min(result['strict_constraints'].values()) <= 0: + raise AssertionError('nonpositive strict outer row') + if result['minimum_margin'] != min(result['margins'].values()): + raise ValueError('wrong minimum margin') + return result +A.assembly = checked_assembly + +bit_path = eng / 'BIT-EXPECTED.json' +bit = json.loads(bit_path.read_bytes()) +bt = bit['nine'] +BH = {int(r): n for r, n in bt['profile'].items()} +BW = bt['bit']['live'] +c = Q(bt['bit_lower']) +b = Q(16834, 10 ** 7) +k = Q(16805, 10 ** 7) +if not 64 * BW - sum((r * n for r, n in BH.items())) == 32 * (8 * bit['word']['v'] - 9 * sum(bit['word']['copy_ranks'])): + raise ValueError('exact arithmetic admission failed') +moments = [] + +def check(name, h, m, w, z, bad): + x = M.moment(h, m, w, z, bad) + y = B.moment(m, w, list(h.items()), z) + if bad: + v = B.moment(m, w, [(1, 32 * m * m * sum(h.values()))], z) + y = (y[0] + v[0] / 10 ** 16, y[1] + v[1] / 10 ** 16) + if not (x[1] < 1 and y[1] < 1): + raise ValueError((name, x, y)) + moments.append({'side': name, 'saving': str(z), 'surcharge': bad, 'mercator_residual': [str(v - 1) for v in x], 'atanh_residual': [str(v - 1) for v in y]}) +check('complex_native', H, 73, W, b, False) +check('complex_conservative', H, 73, W, b, True) +check('bit_retained', BH, 64, BW, c, True) +eta = beta = Q(1, 10 ** 8) +s = min(c, (1 - beta) * b) - Q(1, 10 ** 11) +chain = [Q(384599, 10 ** 10)] +cuts = [] + +def ceil(q): + return -(-q.numerator // q.denominator) +for _ in range(7): + old = chain[-1] + a = (1 - s) * s + s * old + gap = min(s - a, 1 - a - s, 1 - a - s * (1 - old), 1 - s) + if not (old < a < s < c and s < (1 - beta) * b and (0 < gap)): + raise ValueError('exact arithmetic admission failed') + cutoff = max(ceil(36 / gap ** 2), ceil(2 * (4 + 512) / gap)) + if not (cutoff * gap ** 2 >= 36 and cutoff * gap >= 2 * (4 + 512)): + raise ValueError('exact arithmetic admission failed') + chain.append(a) + cuts.append({'saving': str(a), 'gap': str(gap), 'example_constant_bits':512,'example_cutoff_log2': str(cutoff),'universal_cutoff_recipe':'For each sequential fixed C choose integer ell with C < 2^ell; L=max(ceil(36/gap^2),ceil(2*(4+ell)/gap)). These example cutoffs do not bound the actual inherited constants.'}) +bridge = {'semantic': {'C0': {'base': 2, 'exponent': 210000}, 'strict_literal_gap': 1}, 'rows': {'degree_gap': Q(10 ** 6) - Q(51 * 20161, 25)}} +assembly = A.assembly(chain[-1], b, bridge, k, eta=eta, beta=beta) +if not (len(assembly['strict_constraints']) == 47 and min(assembly['strict_constraints'].values()) > 0): + raise ValueError('exact arithmetic admission failed') +controls = [] +for name, kk, ee, bb in [('zero_eta', k, Q(0), beta), ('zero_beta', k, eta, Q(0)), ('unit_beta', k, eta, Q(1)), ('above_balance', assembly['minimum_margin'] + Q(1, 10 ** 12), eta, beta)]: + try: + A.assembly(chain[-1], b, bridge, kk, eta=ee, beta=bb) + except (AssertionError, ZeroDivisionError): + controls.append(name) + else: + raise ValueError(('control accepted', name)) +if not M.moment(H, 73, W, b + Q(1, 10 ** 5), False)[0] > 1: + raise ValueError('exact arithmetic admission failed') +controls.append('supplier_endpoint_too_large') +previous = Q('0.001679754956125544521335670993920702576998311438502580295460327520162864548260325638826305656132844') +if not k > previous + Q(1, 10 ** 7): + raise ValueError('exact arithmetic admission failed') + +def dec(q): + with localcontext() as cx: + cx.prec = 55 + return str(Decimal(q.numerator) / Decimal(q.denominator)) +source_paths = [owners_path, audit_path, trace_path, base / 'PLAN.json', eng / 'moment.py', eng / 'base_two_moment.py', outer_path, bit_path] +result = {'status': 'PASS_RECOUNT_TWO_EXACT_MOMENTS_AND_47_STRICT_OUTER', 'kappa': str(k), 'kappa_decimal': dec(k), 'certified_complex_saving': str(b), 'inherited_bit_saving': str(c), 'increase_over_previous_open_claim': dec(k - previous), 'relative_increase_percent': dec(100 * (k / previous - 1)), 'calls': 72, 'helpers_per_call': 436, 'live': W, 'profile': dict(sorted(H.items())), 'rank_mass': mass, 'deficit': 2560, 'moments': moments, 'bootstrap': cuts, 'outer_parameters': assembly['parameters'], 'outer_constraints': {n: str(v) for n, v in assembly['strict_constraints'].items()}, 'controls_rejected': controls, 'source_sha256': {p.relative_to(campaign).as_posix(): hashlib.sha256(p.read_bytes()).hexdigest() for p in source_paths}, 'scope': 'Exact finite recount and retained conditional outer arithmetic only. Requires independently source-bound native word, physical owner charts/controller, full work/precision/stock invoice and original all-size/analytic/bit assumptions; not full Lean multiplication proof.'} +if args.out: + args.out.parent.mkdir(parents=True, exist_ok=True) + args.out.write_text(json.dumps(result, indent=2, default=str) + '\n') +print(json.dumps({k: v for k, v in result.items() if k in ['status', 'kappa', 'kappa_decimal', 'certified_complex_saving', 'increase_over_previous_open_claim', 'relative_increase_percent', 'profile', 'controls_rejected']})) diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/EXPECTED.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/EXPECTED.json new file mode 100644 index 0000000000..0cb31c95fe --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/EXPECTED.json @@ -0,0 +1,69 @@ +{ + "status": "PASS_ACTUAL_T8_FAMILY_OPERATOR_PARTITION", + "owners": 31392, + "calls": 72, + "helper_count": 436, + "distinct_actual_73_column_charts": 7150, + "parent_factor_count": 387072, + "hosted_factor_count": 254088, + "remainder_factor_count": 132984, + "all_original_child_translations_retained": 16128, + "idle_profile": { + "1": 1724, + "3": 20, + "5": 1469, + "6": 18, + "7": 494, + "32": 11, + "39": 7, + "48": 2493 + }, + "complete_profile": { + "1": 176252, + "2": 30096, + "3": 30908, + "4": 10152, + "5": 4493, + "6": 18, + "7": 494, + "8": 576, + "32": 11, + "39": 7, + "48": 2493 + }, + "rank_mass": 520704, + "native_ABI_rechecks": [ + { + "orientation": "forward", + "helpers": 436, + "unique_literal_operator_checks": 26, + "trace_sha256": "a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee", + "native_payload_sha256": "c8c8152fdab3080402a9b1d6e89ba831045bfc575bb1ef2a18a7157957fd62b4", + "wrapper_sha256": "e8a95fb4005f8cbaaf81055678a14c4b0bf648fb72eb0593ac7439d9a59dc6ce" + }, + { + "orientation": "backward", + "helpers": 436, + "unique_literal_operator_checks": 26, + "trace_sha256": "fc965c4a041f26960268b78ba526b9cca13cd01d713a843f31fe4a82c4e08334", + "native_payload_sha256": "e5214a0780f58f407a6163eef8e99ccacfd77741e1897a3725941616e5ba6c05", + "wrapper_sha256": "75e530e70dafe699b4316fda6cda249197553d2cdf317bff24c7f5f992a896bd" + } + ], + "mutations_rejected": [ + "negative_child_direction", + "cross_family_axis_swap_with_same_global_axis_counts", + "wrong_global_scalar_phase" + ], + "source_sha256": { + "parent": "025448abdf2012d327e50a009084e1501690d05e650300d2982836aa0350ae00", + "phases": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa", + "plan": "7b3122026e82b9d9e964d2a50c7bffdc976ae3dfea64bb98b2fdad5cfbdbaa20", + "owners": "892b49f97d4caa7e92918262ea3a8cfa8cbf273c92d574ae53bb89a61ee5e847", + "forward_trace": "a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee", + "backward_trace": "fc965c4a041f26960268b78ba526b9cca13cd01d713a843f31fe4a82c4e08334", + "native_engine": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38" + }, + "actual_factors_sha256": "81289c868dc9672572154e3cdebddc43241491f16f06353512e7c4b9a18f4881", + "scope": "Exact finite family-tagged Gaussian factor/actual chronology/chart/ABI composition. Shares public local signed-Pauli and exact-Gaussian checker with native audits. No all-size compiler, precision/tape, stock/invoice or analytic transfer is established here." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/PROOF.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/PROOF.md new file mode 100644 index 0000000000..1e959be0f9 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/PROOF.md @@ -0,0 +1,11 @@ +# Exact operator partition on physical families + +Run `python -I -O -B independent/verify.py` from the contribution directory. Only Python's standard library is required. The source package is read-only; generated charts and receipts go to a temporary directory. To retain them, pass `--work` with a nonexistent directory outside the source package. `SOURCES.json` pins every consumed source and native checker. `EXPECTED.json` freezes the complete deterministic receipt. The literal input parent is the T8, 72-call parent retained by the public PR1005 owner-repair lineage, not the incompatible T74 parent. + +Write `X_v f(x)=f(x+v)` and `C_v=((1+i)I+(1-i)X_v)/2`. The algebraic identities in `../lean/Factors.lean` prove that these full Gaussian operators commute on an additive commutative address space, including their scalar phases. They also prove that a permutation of instructions preserving both physical family and axis preserves the family-wise operator, and that address translations commute with these products. Thus a literal factor partition within one untouched family interval is a sufficient equality certificate. A global bag of axes that forgets the physical family is insufficient. + +This checker reconstructs the actual parent chronology, including both bridge reads and bridge writes. Each replica preserves the parent's source event sequence. Every complete helper invocation lies inside its donor child's untouched interval. Every helper is assigned to a distinct physical family outside that invocation's 224 source and target data families. The 73-column orthogonal charts are derived from the parent's actual port normals. Their first `w` columns are the literal assigned pending axes, with the remaining local axes treated as identity spectators. Both orientations' literal `R^-1` and `L^-1` wrappers are rechecked against their actual native payloads, including exact global phase: `L^-1 (T_F T_E^-1) R^-1 = C_w tensor I_(9-w)`. Data chart normals and source/target directions are checked against the source parent. + +The audit partitions all 387,072 positive parent idle-child factors into 254,088 hosted factors and 132,984 literal remaining factors, retaining every physical family, source child, axis, and positive direction. All 16,128 original positive-idle-child record translations are explicitly retained, even when the remaining Gaussian rank is zero. The rebuilt paid profile has rank mass 520,704. Mutation controls reject a negative child direction, a cross-family axis swap that preserves the untagged global axis bag, and a wrong scalar phase. These controls specifically address the family-operator failure of the withdrawn cross-family axis-swap proposals. + +The factor partition and chronology implementation imports no hosting constructor or owner emitter. Its native Gaussian interpreter is the same retained public signed-Pauli/exact-Gaussian engine used by the local native audit; this is an independent assembly audit, not a third independent native implementation. This finite certificate establishes the stated operator partition. It does not establish an all-size compiler, precision or tape complexity, the physical invoice, or the analytic transfer to multiplication; those are separate parts of the package and the project's inherited hypotheses. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/SOURCES.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/SOURCES.json new file mode 100644 index 0000000000..392efef274 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/SOURCES.json @@ -0,0 +1,18 @@ +{ + "independent/inputs/T8-PARENT.json.gz": "025448abdf2012d327e50a009084e1501690d05e650300d2982836aa0350ae00", + "independent/audit.py": "65ac584ba450d5fad1dfef818bad2e098dba5c974bc9cc8216efa192ceb3c82b", + "independent/verify.py": "a39182cbefc8ee6ac38f8147e437d3f93407328bc9ab2424b2a694dc8eed12d2", + "independent/EXPECTED.json": "70c921be4d0292c22c045090382d7f08fc84e851dce9969bdc16573c6bc8b1f3", + "lean/Factors.lean": "5073b72b68037e401be2f48f104f122ddd5193bc8f349cb7d4a10c906ddc5cf5", + "owners/OWNERS.json.gz": "892b49f97d4caa7e92918262ea3a8cfa8cbf273c92d574ae53bb89a61ee5e847", + "owners/inputs/DATA-PHASES.json": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa", + "owners/inputs/PLAN.json": "7b3122026e82b9d9e964d2a50c7bffdc976ae3dfea64bb98b2fdad5cfbdbaa20", + "local/data/forward.json": "a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee", + "local/data/backward.json": "fc965c4a041f26960268b78ba526b9cca13cd01d713a843f31fe4a82c4e08334", + "local/data/native-forward.json.gz": "5ee6191bde203a43c429de681cd405f45932b6727f01623673b223072bfb0e15", + "local/data/native-backward.json.gz": "ba58e5e557996a15470259e5b01a7db164151b878fdac76900d57fe94d4626f4", + "local/data/wrappers-forward.json": "e8a95fb4005f8cbaaf81055678a14c4b0bf648fb72eb0593ac7439d9a59dc6ce", + "local/data/wrappers-backward.json": "75e530e70dafe699b4316fda6cda249197553d2cdf317bff24c7f5f992a896bd", + "local/vendor/native_verify.py": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38", + "local/vendor/clifford_exact.py": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/audit.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/audit.py new file mode 100644 index 0000000000..502dafdf43 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/independent/audit.py @@ -0,0 +1,208 @@ +"""Independent T8 family-tag and Gaussian-factor audit of literal owners. + +No optimizer/emitter/hosting constructor is imported. Window boundaries come +from the pinned parent program including bridge reads and writes. Local ABI +normalization is rechecked using the separately implemented public signed- +Pauli/exact-Gaussian interpreter; this local engine is shared with the native +audit and is explicitly not a third independent native implementation. + +For C_v=((1+i)I+(1-i)X_v)/2, all shifts X_v commute on one physical family. +The literal factor multiset, including physical family and positive direction, +therefore suffices for the pending-child operator equality. Untagged global +axis counts do not suffice and are rejected by a cross-family mutation. +""" +import argparse,collections,gzip,hashlib,importlib.util,json,sys,time +from pathlib import Path +sys.dont_write_bytecode=True + +def need(ok,message): + if not ok:raise ValueError(message) +def sha(path):return hashlib.sha256(Path(path).read_bytes()).hexdigest() +def read(path): + p=Path(path);data=p.read_bytes() + return json.loads(gzip.decompress(data) if p.suffix=='.gz' else data) +def packed(value):return json.dumps(value,sort_keys=True,separators=(',',':')).encode() +def tuples(value):return tuple(map(tuples,value)) if isinstance(value,list) else value +def load(name,path): + spec=importlib.util.spec_from_file_location(name,path);module=importlib.util.module_from_spec(spec) + sys.modules[name]=module;spec.loader.exec_module(module);return module + +def orthogonal_basis(q): + z=next(i for i in range(9) if not(q>>i&1));u=q^(1<>i&1 else 0) for i in indices]+[1<=112) + need((adapter['role'],adapter['position'])==(('source','entry') if source else ('target','exit')),'actual data adapter direction and endpoint') + need(adapter['recipe']['rank']==0,'all actual data chart adapters are monomial') + ranks={(x['child'],x['port']):x['rank'] for x in parent['idle_children']} + need(len(ranks)==len(parent['idle_children'])==2016,'all actual positive-rank parent children') + for (child,port),rank in ranks.items(): + record=phases['recipes'][port]['idle_children'][child];axes=record['orthogonal_axes'] + need(rank==record['rank']==len(axes),'actual parent rank/axes equality') + need(len(set(axes))==rank and all(v in bases[port] for v in axes),'literal pending axes in actual orthogonal chart') + used=collections.defaultdict(set);by_call=collections.defaultdict(dict);hosted_factors=collections.Counter();charts={};roles=[] + for row in owners: + time_index=row['time'];stage,replica=order[time_index];helper=row['helper'];width=widths[helper] + need((row['stage'],row['replica'],row['width'])==(stage,replica,width),'actual role identity and width') + need(helper not in by_call[time_index],'one actual helper role per invocation') + owner_replica,child,port=row['owner_replica'],row['child'],row['port'];pair,side,event=tags[child] + need(0<=owner_replica<8 and 0<=port<112,'actual owner bounds') + physical=owner_replica*896+(2*pair+side)*112+port + need(row['family']==physical,'actual parent physical family tag') + need(row['copy']==owner_replica*112+port,'canonical old copy ABI metadata') + lower,upper=windows[owner_replica,child] + need(lower a * f x + b * f (x + v) + +theorem weightedShift_commute [AddCommMonoid G] [CommSemiring R] + (a b c d : R) (v w : G) (f : G → R) : + weightedShift a b v (weightedShift c d w f) = + weightedShift c d w (weightedShift a b v f) := by + funext x + simp only [weightedShift] + rw [show (x + v) + w = (x + w) + v by ac_rfl] + ring + +/-- The full positive native Gaussian operator, including its scalar phase. -/ +def gaussianStep [AddCommMonoid G] (v : G) (f : G → ℂ) : G → ℂ := + weightedShift ((1 + Complex.I) / 2) ((1 - Complex.I) / 2) v f + +theorem gaussianStep_commute [AddCommMonoid G] + (v w : G) (f : G → ℂ) : + gaussianStep v (gaussianStep w f) = gaussianStep w (gaussianStep v f) := + weightedShift_commute _ _ _ _ _ _ _ + +def gaussianWord [AddCommMonoid G] (word : List G) (f : G → ℂ) : G → ℂ := + word.foldr gaussianStep f + +theorem gaussianWord_append [AddCommMonoid G] + (xs ys : List G) (f : G → ℂ) : + gaussianWord (xs ++ ys) f = gaussianWord xs (gaussianWord ys f) := by + simp [gaussianWord, List.foldr_append] + +theorem gaussianWord_perm [AddCommMonoid G] + {xs ys : List G} (h : xs.Perm ys) (f : G → ℂ) : + gaussianWord xs f = gaussianWord ys f := by + induction h with + | nil => rfl + | cons a h ih => + change gaussianStep a (gaussianWord _ f) = gaussianStep a (gaussianWord _ f) + rw [ih] + | swap a b l => + change gaussianStep _ (gaussianStep _ _) = gaussianStep _ (gaussianStep _ _) + apply gaussianStep_commute + | trans h₁ h₂ ih₁ ih₂ => exact ih₁.trans ih₂ + +/-- Every factor acts on its named physical family and leaves all others fixed. -/ +def familyStep [DecidableEq K] [AddCommMonoid G] + (instruction : K × G) (f : K → G → ℂ) : K → G → ℂ := + fun k => if k = instruction.1 then gaussianStep instruction.2 (f k) else f k + +theorem familyStep_commute [DecidableEq K] [AddCommMonoid G] + (a b : K × G) (f : K → G → ℂ) : + familyStep a (familyStep b f) = familyStep b (familyStep a f) := by + funext k + by_cases ha : k = a.1 + · by_cases hb : k = b.1 + · have hab : a.1 = b.1 := ha.symm.trans hb + simpa [familyStep, ha, hab] using gaussianStep_commute a.2 b.2 (f k) + · have hab : a.1 ≠ b.1 := fun h => hb (ha.trans h) + simp [familyStep, ha, hab] + · by_cases hb : k = b.1 + · have hba : b.1 ≠ a.1 := fun h => ha (hb.trans h) + simp [familyStep, hb, hba] + · simp [familyStep, ha, hb] + +def familyWord [DecidableEq K] [AddCommMonoid G] + (word : List (K × G)) (f : K → G → ℂ) : K → G → ℂ := + word.foldr familyStep f + +/-- A permutation certificate must retain the family as well as the axis. -/ +theorem familyWord_perm [DecidableEq K] [AddCommMonoid G] + {xs ys : List (K × G)} (h : xs.Perm ys) (f : K → G → ℂ) : + familyWord xs f = familyWord ys f := by + induction h with + | nil => rfl + | cons a h ih => + change familyStep a (familyWord _ f) = familyStep a (familyWord _ f) + rw [ih] + | swap a b l => + change familyStep _ (familyStep _ _) = familyStep _ (familyStep _ _) + apply familyStep_commute + | trans h₁ h₂ ih₁ ih₂ => exact ih₁.trans ih₂ + +def addressShift [AddCommMonoid G] (q : G) (f : G → ℂ) : G → ℂ := + fun x => f (x + q) + +theorem gaussianStep_addressShift [AddCommMonoid G] + (v q : G) (f : G → ℂ) : + gaussianStep v (addressShift q f) = addressShift q (gaussianStep v f) := by + funext x + simp only [gaussianStep, weightedShift, addressShift] + rw [show (x + v) + q = (x + q) + v by ac_rfl] + +theorem gaussianWord_addressShift [AddCommMonoid G] + (word : List G) (q : G) (f : G → ℂ) : + gaussianWord word (addressShift q f) = addressShift q (gaussianWord word f) := by + induction word with + | nil => rfl + | cons v vs ih => + change gaussianStep v (gaussianWord vs (addressShift q f)) = + addressShift q (gaussianStep v (gaussianWord vs f)) + rw [ih, gaussianStep_addressShift] + +end + +end SelectiveBorrowing diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/lean/Feedback.lean b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/lean/Feedback.lean new file mode 100644 index 0000000000..59d0ebae71 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/lean/Feedback.lean @@ -0,0 +1,39 @@ +import Mathlib.Tactic.Linarith +import Mathlib.Data.Real.Basic + +/-! Sufficient all-cut feedback bounds for the selective-borrowing invoice. +These lemmas do not assert that a particular implementation meets the defect +contract; the independently recounted instruction envelope supplies that data. +-/ + +namespace IntegerMultBounds.SelectiveBorrowing + +theorem absorb_feedback (error coefficient budget : ℝ) + (herror : 0 ≤ error) (hsmall : coefficient ≤ 1 / 2) + (hfeedback : error ≤ coefficient * (budget + error)) : + error ≤ 2 * coefficient * budget := by + have h := mul_nonneg (sub_nonneg.mpr hsmall) herror + nlinarith + +theorem relative_error_of_feedback (error amplification defect unit initial target : ℝ) + (herror : 0 ≤ error) (hinitial : 0 ≤ initial) + (hsmall : defect * unit ≤ 1 / 2) + (hbudget : 2 * defect * unit * amplification ≤ target) + (hfeedback : error ≤ defect * unit * (amplification * initial + error)) : + error ≤ target * initial := by + calc + error ≤ 2 * (defect * unit) * (amplification * initial) := + absorb_feedback error (defect * unit) (amplification * initial) herror hsmall hfeedback + _ = (2 * defect * unit * amplification) * initial := by simp only [mul_assoc] + _ ≤ target * initial := mul_le_mul_of_nonneg_right hbudget hinitial + +theorem owner_stock_reserve (oldHelpers newHelpers cleanup : ℕ) + (hreserve : newHelpers ≤ oldHelpers) : + newHelpers + cleanup ≤ oldHelpers + cleanup := + Nat.add_le_add_right hreserve cleanup + +end IntegerMultBounds.SelectiveBorrowing + +#print axioms IntegerMultBounds.SelectiveBorrowing.absorb_feedback +#print axioms IntegerMultBounds.SelectiveBorrowing.relative_error_of_feedback +#print axioms IntegerMultBounds.SelectiveBorrowing.owner_stock_reserve diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/lean/README.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/lean/README.md new file mode 100644 index 0000000000..3fd9b9007f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/lean/README.md @@ -0,0 +1,17 @@ +# Optional Mathlib-only universal lemmas + +Checked with Lean 4.33.0 and Mathlibv 4.33.0 in fresh kernel subprocesses, +followed by independent axiom inspection. Every theorem uses at most +propext, Classical.choice and Quot.sound. CHECKS.json binds the exact sources. + +In a compatible Mathlib checkout, run `lake env lean /path/to/Factors.lean` +and `lake env lean /path/to/Feedback.lean`. Factors proves weighted Gaussian +shifts commute and words preserve their action under permutations that retain +physical family tags; address translations commute too. Feedback proves a +sufficient contraction inequality and a reserve monotonicity inequality. + +These eleven universal lemmas are useful sufficient algebraic facts. They do +not certify a particular owner table, reflect the Python computations, prove +the retained all-size compiler, or establish the multiplication bound in Lean. +No file imports a personal Mathproof module. To reproduce the full axiom +inspection, append `#print axioms` for each qualified theorem in CHECKS.json. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/SOURCE-PINS.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/SOURCE-PINS.json new file mode 100644 index 0000000000..452c13187e --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/SOURCE-PINS.json @@ -0,0 +1,119 @@ +{ + "status": "FROZEN_PORTABLE_SOURCES_AND_NATIVE_PROGRAMS", + "files": [ + { + "path": "check_trace.py", + "sha256": "895a02ef318031cb55b71a36f234a2e9808029363fad2c1a8aa9fc3069fb33d6" + }, + { + "path": "data/backward.json", + "sha256": "fc965c4a041f26960268b78ba526b9cca13cd01d713a843f31fe4a82c4e08334" + }, + { + "path": "data/expected-local.json", + "sha256": "90f33cff012d98cc467dba95c49fa195931cb43539e78b78d0a0647014bfeb9e" + }, + { + "path": 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{ + "path": "data/wrappers-forward.json", + "sha256": "e8a95fb4005f8cbaaf81055678a14c4b0bf648fb72eb0593ac7439d9a59dc6ce" + }, + { + "path": "inputs/selected-block-certificate.json", + "sha256": "d3ada9ac84a74f72ce9364dd47ee2ab9f02d1a6513df135e81dad11d0e6c6958" + }, + { + "path": "inputs/selected-complex-word.json.gz", + "sha256": "16bc0ebb04b4be6c0ade74badc6a0c0cdba4215eda86e4ff9f13abf8ecebd219" + }, + { + "path": "inputs/WITNESS.json", + "sha256": "0a696e6243ba611eec064d0a3bb28fa94ca263d97eec22830f83fabf2e96616b" + }, + { + "path": "invoice.py", + "sha256": "9f34aca05f3cabb95cfb58565246b690ca7093e786f5e14ce3b0d660e62186c8" + }, + { + "path": "native.py", + "sha256": "7d70f7396d1056af1fc8cd04f0bd00ea4c68c3e05b40fd6896130bfa10e2e43a" + }, + { + "path": "precision.py", + "sha256": "4cb9fb27f73f21cf4334f61edee0b64e5a243d05089fa422ae5b714b5e7abffc" + }, + { + "path": "rebuild.py", + "sha256": "3861a38c7216d8df8f974af864e029af352d15924b473f42330d74139886d9ac" + }, + { + "path": 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"path": "verify.py", + "sha256": "2c8c1ff89cbcfe776344e8aabbaec7902f3a71cd8cac868ddf84ee9211a6bde5" + } + ], + "source_pr2226_head": "dacaf3d81f1936bb55b8454b1244e5d014272754", + "source_pr1991_native_compiler_sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/check_trace.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/check_trace.py new file mode 100644 index 0000000000..e60a46d354 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/check_trace.py @@ -0,0 +1,125 @@ +"""Independent exact row replay of the literal complete forward trace.""" +import argparse +import hashlib +import json +from collections import Counter +from fractions import Fraction +from pathlib import Path + +HERE = Path(__file__).resolve().parent + +def need(condition, message): + if not condition: + raise ValueError(message) + +def span(vectors): + result = {0} + for v in vectors: + result.update(x ^ v for x in list(result)) + return frozenset(result) + +def run(data): + active = data['active_registers'] + values = {r: {(r, ()): Fraction(1)} for r in active} + frames = {r: span(data['endpoints'][str(r)]['initial']) for r in active} + private = set() + profile = Counter() + maximum_row_l1 = Fraction(1) + maximum_at = None + def add_to(d, s, coefficient): + need(d != s, 'self addition') + need(frames[d] == frames[s], 'literal ADD common chart') + for k, value in list(values[s].items()): + new = values[d].get(k, 0) + coefficient * value + if new: + values[d][k] = new + else: + values[d].pop(k, None) + def warp(r, symbol): + result = {} + for (var, word), coefficient in values[r].items(): + transformed = word[1:] if word and word[0] == -symbol else (symbol,) + word + key = var, transformed + result[key] = result.get(key, 0) + coefficient + values[r] = {k: q for k, q in result.items() if q} + for index, e in enumerate(data['events']): + kind = e[0] + if kind == 'transport': + _, r, old, new, paid = e + need(frames[r] == span(old) and span(old) < span(new) or + r in private and frames[r] == span(old) and span(new) < span(old), + ('nested physical transport', index)) + need(paid == abs(len(new) - len(old)), 'actual positive dimension gap') + frames[r] = span(new) + profile[paid] += 1 + elif kind == 'add': + _, d, s, q, common = e + need(frames[d] == span(common) == frames[s], ('ADD chart', index)) + add_to(d, s, Fraction(q)) + elif kind == 'scale': + _, r, q, common = e + need(frames[r] == span(common), 'SCALE chart') + q = Fraction(q) + need(q != 0, 'invertible SCALE') + values[r] = {k: q * value for k, value in values[r].items()} + elif kind == 'swap': + _, d, s, common = e + need(frames[d] == frames[s] == span(common), 'SWAP chart') + values[d], values[s] = values[s], values[d] + elif kind in ('logical_monomial', 'dual_monomial'): + _, r, symbol, common = e + need(frames[r] == span(common), 'monomial chart') + warp(r, symbol) + elif kind in ('handoff', 'dual_handoff'): + _, r, old, new, symbol = e + need(frames[r] == span(old) and len(old) == len(new), 'handoff chart') + warp(r, symbol) + frames[r] = span(new) + elif kind == 'copy': + _, r, source, common = e + need(r not in values or not values[r], 'owned COPY zero guard') + need(frames[source] == span(common), 'COPY source frame') + values[r] = dict(values[source]) + frames[r] = span(common) + private.add(r) + elif kind == 'clear': + _, r, common = e + need(r in private and frames[r] == span(common), 'owned CLEAR guard') + values[r] = {} + elif kind == 'zero_buffer': + _, r, common, restored_role = e + need(r not in values and restored_role in active, 'fresh zero buffer') + values[r] = {} + frames[r] = span(common) + private.add(r) + elif kind == 'zero_reframe': + _, r, common = e + need(r in private and not values[r], 'only a proved zero array may be reframed freely') + frames[r] = span(common) + else: + raise ValueError(('unknown literal event', e)) + touched = (e[1:3] if kind in ('add', 'swap') else e[1:2]) + for r in touched: + if r not in values: + continue + magnitude = sum(abs(q) for q in values[r].values()) + if magnitude > maximum_row_l1: + maximum_row_l1, maximum_at = magnitude, [index, r] + for r in active: + expected = {(r, ()): Fraction(1)} + if data.get('orientation') == 'backward': + if r < 112: + expected[r + 112, ()] = Fraction(-1) + elif 112 <= r < 224: + expected[r - 112, ()] = Fraction(1) + need(values[r] == expected, ('complete endpoint independent column', r)) + need(frames[r] == span(data['endpoints'][str(r)]['final']), 'endpoint chart') + need(all(not values[r] for r in private), 'every owned private buffer returned zero') + need({str(k): v for k, v in profile.items()} == data['local_profile'], 'literal paid profile') + return dict(status='PASS_COMPLETE_' + data.get('orientation', 'forward').upper() + '_LITERAL_TRACE', independent_input_columns=len(active), + events=len(data['events']), private_fields=len(private), + helper_width_inventory=dict(sorted(Counter(data['helper_widths'].values()).items())), + paid_profile=dict(sorted(profile.items())), + rank_mass=sum(r * count for r, count in profile.items()), + exact_free_operator_prefix_row_l1=str(maximum_row_l1), + prefix_peak_event_and_register=maximum_at) diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/backward.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/backward.json new file mode 100644 index 0000000000..ee9b68d849 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/backward.json @@ -0,0 +1,312992 @@ +{ + "status": "EXPORTED_COMPLETE_BACKWARD_REFLECTED_ADJOINT_NATIVE_PENDING", + "orientation": "backward", + "source_forward_sha256": 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76, + "scale": 1312, + "add": 16340, + "dual_monomial": 44, + "swap": 4, + "clear": 91, + "transport": 3462, + "dual_handoff": 44, + "zero_reframe": 8 + }, + "dual_logical_operator": "J_i = F K_i^dagger F^(-1)", + "expected_data_map": "X_i -= Y_i; Y_i unchanged" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/expected-local.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/expected-local.json new file mode 100644 index 0000000000..2afa925b69 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/expected-local.json @@ -0,0 +1,129 @@ +{ + "status": "PASS_COMPLETE_TWO_ORIENTATION_LOCAL_NATIVE_VARIANT", + "reports": { + "forward": { + "status": "PASS_COMPLETE_FORWARD_LITERAL_TRACE", + "independent_input_columns": 660, + "events": 21065, + "private_fields": 76, + "helper_width_inventory": { + "4": 40, + "5": 4, + "7": 27, + 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This is neither a rounded-child implementation nor selected global stock/row or all-size admission." + } + }, + "backward": { + "status": "PASS_COMPLETE_BACKWARD_LITERAL_TRACE", + "independent_input_columns": 660, + "events": 21381, + "private_fields": 76, + "helper_width_inventory": { + "4": 40, + "5": 4, + "7": 27, + "8": 125, + "9": 240 + }, + "paid_profile": { + "1": 2424, + "2": 418, + "3": 429, + "4": 141, + "5": 42, + "8": 8 + }, + "rank_mass": 5385, + "exact_free_operator_prefix_row_l1": "101", + "prefix_peak_event_and_register": [ + 11367, + 45 + ], + "wrappers": { + "helper_wrappers": 436, + "intrinsic_data_adapters": 224 + }, + "native_jobs": 2996, + "precision": { + "status": "PASS_ALL_CUT_NONCANCELLATION_ENVELOPE", + "instructions_after_horner_and_swap_expansion": 85369, + "ideal_prefix_P": "15709105296279064519/2", + "uniform_completed_instruction_defect_K": "105657879540020779682", + "P_binary_upper": 63, + "K_binary_upper": 67, + "local_sufficient_guard_bits": 133, + "ideal_peak": [ + 83075, + 319 + ], + "error_peak": [ + 83075, + 319 + ], + "completed_instruction_defect_contract": "For each completed recursive unitary/COPY step, Hilbert defect <= u times maximum current stream norm. Each real arithmetic primitive has defect <= u times its row norm times that maximum; the envelope charges a factor two when combining the real and imaginary components. All compound ADD coefficients are expanded through an actual owned Horner work row; SCALE and SWAP are expanded to their real primitives. Exact clear/zero allocation writes zero.", + "sufficient_implication": "For every b >= 0 and initial maximum stream norm B >= 0, u <= 2^(-b-guard) gives every-cut error <= B*2^(-b), strictly when B > 0. At B=0 the stated relative-defect contract forces exact zero.", + "proof": "The nonnegative recurrence bounds every ideal state by P*B and every sequence of uniform unit defects by K at every cut. For E the maximum actual error, linear propagation gives E <= K*u*(P*B+E). Since K*u < 1/2, E <= K*u*P*B/(1-K*u). The recorded dyadic guard makes this less than B*2^(-b).", + "scope": "Finite local conditional precision implication only. This is neither a rounded-child implementation nor selected global stock/row or all-size admission." + } + } + }, + "source_hashes": { + "forward_trace": "a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee", + "forward_native_payload": "c8c8152fdab3080402a9b1d6e89ba831045bfc575bb1ef2a18a7157957fd62b4", + "backward_trace": "fc965c4a041f26960268b78ba526b9cca13cd01d713a843f31fe4a82c4e08334", + "backward_native_payload": "e5214a0780f58f407a6163eef8e99ccacfd77741e1897a3725941616e5ba6c05" + }, + "negative_controls": [ + "forward_transport_phase", + "forward_helper_residual_phase", + "wrong_last_target_coefficient", + "missing_actual_reference_change", + "missing_native_occurrence_binding", + "backward_transport_phase", + "backward_helper_residual_phase" + ], + "source_compiler_sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962", + "independent_native_interpreter_sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38", + "scope": "Exact finite local words, arbitrary dirty logical restoration, owned-zero scratch and all actual native phase bindings. Whole-call owners, physical finite invoice, rounded child admission, tape and all-size transfer remain separate." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/finite-invoice.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/finite-invoice.json new file mode 100644 index 0000000000..5f41d75e65 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/finite-invoice.json @@ -0,0 +1,154 @@ +{ + "status": "PASS_FRESH_CONSERVATIVE_FINITE_INVOICE_UNDER_RETAINED_INTERFACES", + "local": { + "forward": { + "expanded_event_counts": { + "scale": 1092, + "add": 16344, + "transport": 3462, + "copy": 8, + "clear": 8, + "handoff": 44, + "zero_buffer": 75, + "logical_monomial": 44 + }, + "scalar_macros": 17603, + "real_component_primitives": 162954, + "coefficient_work_primitives": 79864, + "extra_exact_private_postlude_clears": 76, + "conservative_macro_envelope": 21153, + "precision": { + "status": "PASS_ALL_CUT_NONCANCELLATION_ENVELOPE", + "instructions_after_horner_and_swap_expansion": 85029, + "ideal_prefix_P": "4775934987978459611/2", + "uniform_completed_instruction_defect_K": "19304968969869700845", + "P_binary_upper": 62, + "K_binary_upper": 65, + "local_sufficient_guard_bits": 130, + "ideal_peak": [ + 83598, + 313 + ], + "error_peak": [ + 83598, + 313 + ], + "completed_instruction_defect_contract": "For each completed recursive unitary/COPY step, Hilbert defect <= u times maximum current stream norm. Each real arithmetic primitive has defect <= u times its row norm times that maximum; the envelope charges a factor two when combining the real and imaginary components. All compound ADD coefficients are expanded through an actual owned Horner work row; SCALE and SWAP are expanded to their real primitives. Exact clear/zero allocation writes zero.", + "sufficient_implication": "For every b >= 0 and initial maximum stream norm B >= 0, u <= 2^(-b-guard) gives every-cut error <= B*2^(-b), strictly when B > 0. At B=0 the stated relative-defect contract forces exact zero.", + "proof": "The nonnegative recurrence bounds every ideal state by P*B and every sequence of uniform unit defects by K at every cut. For E the maximum actual error, linear propagation gives E <= K*u*(P*B+E). Since K*u < 1/2, E <= K*u*P*B/(1-K*u). The recorded dyadic guard makes this less than B*2^(-b).", + "scope": "Finite local conditional precision implication only. 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"15709105296279064519/2", + "uniform_completed_instruction_defect_K": "105657879540020779682", + "P_binary_upper": 63, + "K_binary_upper": 67, + "local_sufficient_guard_bits": 133, + "ideal_peak": [ + 83075, + 319 + ], + "error_peak": [ + 83075, + 319 + ], + "completed_instruction_defect_contract": "For each completed recursive unitary/COPY step, Hilbert defect <= u times maximum current stream norm. Each real arithmetic primitive has defect <= u times its row norm times that maximum; the envelope charges a factor two when combining the real and imaginary components. All compound ADD coefficients are expanded through an actual owned Horner work row; SCALE and SWAP are expanded to their real primitives. Exact clear/zero allocation writes zero.", + "sufficient_implication": "For every b >= 0 and initial maximum stream norm B >= 0, u <= 2^(-b-guard) gives every-cut error <= B*2^(-b), strictly when B > 0. At B=0 the stated relative-defect contract forces exact zero.", + "proof": "The nonnegative recurrence bounds every ideal state by P*B and every sequence of uniform unit defects by K at every cut. For E the maximum actual error, linear propagation gives E <= K*u*(P*B+E). Since K*u < 1/2, E <= K*u*P*B/(1-K*u). The recorded dyadic guard makes this less than B*2^(-b).", + "scope": "Finite local conditional precision implication only. This is neither a rounded-child implementation nor selected global stock/row or all-size admission.", + "trace_sha256": "fc965c4a041f26960268b78ba526b9cca13cd01d713a843f31fe4a82c4e08334" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/precision-forward.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/precision-forward.json new file mode 100644 index 0000000000..86c3310742 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/precision-forward.json @@ -0,0 +1,22 @@ +{ + "status": "PASS_ALL_CUT_NONCANCELLATION_ENVELOPE", + "instructions_after_horner_and_swap_expansion": 85029, + "ideal_prefix_P": "4775934987978459611/2", + "uniform_completed_instruction_defect_K": "19304968969869700845", + "P_binary_upper": 62, + "K_binary_upper": 65, + "local_sufficient_guard_bits": 130, + "ideal_peak": [ + 83598, + 313 + ], + "error_peak": [ + 83598, + 313 + ], + "completed_instruction_defect_contract": "For each completed recursive unitary/COPY step, Hilbert defect <= u times maximum current stream norm. Each real arithmetic primitive has defect <= u times its row norm times that maximum; the envelope charges a factor two when combining the real and imaginary components. All compound ADD coefficients are expanded through an actual owned Horner work row; SCALE and SWAP are expanded to their real primitives. Exact clear/zero allocation writes zero.", + "sufficient_implication": "For every b >= 0 and initial maximum stream norm B >= 0, u <= 2^(-b-guard) gives every-cut error <= B*2^(-b), strictly when B > 0. At B=0 the stated relative-defect contract forces exact zero.", + "proof": "The nonnegative recurrence bounds every ideal state by P*B and every sequence of uniform unit defects by K at every cut. For E the maximum actual error, linear propagation gives E <= K*u*(P*B+E). Since K*u < 1/2, E <= K*u*P*B/(1-K*u). The recorded dyadic guard makes this less than B*2^(-b).", + "scope": "Finite local conditional precision implication only. This is neither a rounded-child implementation nor selected global stock/row or all-size admission.", + "trace_sha256": "a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/wrappers-backward.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/wrappers-backward.json new file mode 100644 index 0000000000..42ab6d7c82 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/wrappers-backward.json @@ -0,0 +1,60767 @@ +{ + "status": "PASS_BACKWARD_ENDPOINT_AND_SPECTATOR_WRAPPERS", + "trace_sha256": "fc965c4a041f26960268b78ba526b9cca13cd01d713a843f31fe4a82c4e08334", + "native_payload_sha256": "e5214a0780f58f407a6163eef8e99ccacfd77741e1897a3725941616e5ba6c05", + "helper_count": 436, + "data_adapter_count": 224, + "helpers": { + "224": { + "width": 9, + "initial": [], + 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phase.", + "scope": "Exact finite local ABI only. Placement into an admitted global owner chart and all implementation costs remain separately charged." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/wrappers-forward.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/wrappers-forward.json new file mode 100644 index 0000000000..3fc17f68f4 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/data/wrappers-forward.json @@ -0,0 +1,60818 @@ +{ + "status": "PASS_FORWARD_ENDPOINT_AND_SPECTATOR_WRAPPERS", + "trace_sha256": "a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee", + "native_payload_sha256": "c8c8152fdab3080402a9b1d6e89ba831045bfc575bb1ef2a18a7157957fd62b4", + "helper_count": 436, + "data_adapter_count": 224, + "helpers": { + "224": { + "width": 9, + "initial": [], + "final": [ + 256, + 128, + 64, + 32, + 16, + 8, + 4, + 2, + 1 + ], + 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Placement into an admitted global owner chart and all implementation costs remain separately charged." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/inputs/WITNESS.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/inputs/WITNESS.json new file mode 100644 index 0000000000..2a8a533287 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/inputs/WITNESS.json @@ -0,0 +1,1866 @@ +{ + "pairs": [ + { + "register": 597, + "base": 592, + "common_basis": [ + 320, + 192, + 45, + 30 + ], + "dimension": 4, + "deadline": 1054, + "deleted_echo_indices": [], + "low": false, + "borrow_from": 243, + "retirement_B_event": -1, + "retirement_frame": 66, + "retirement_dimension": 4, + "old_final_dimension": 9, + "handoff_B": 1054, + "handoff_basis": [ + 320, + 192, + 45, + 30 + ], + "handoff_paid_rank": 0, + "anticipate": false + }, + { + "register": 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b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/invoice.py @@ -0,0 +1,148 @@ +"""Fresh conservative finite invoice from actual local words and owner table. + +Retains the existing m=73 chart compiler and finite cover/row conventions, but +recounts the new program, all scratch and exact fine-grid magnitude/depth guards. +The separate optional rounding envelope is not an inherited child premise. +""" +import argparse +import gzip +import hashlib +import json +import sys +from collections import Counter,defaultdict +from fractions import Fraction as Q +from pathlib import Path + +HERE=Path(__file__).resolve().parent +sys.path.insert(0,str(HERE)) +from precision import calculate,expanded + +def need(condition,label): + if not condition:raise ValueError(label) + +def scalar_count(trace): + counts=Counter();real=work=0;macros=0 + for event in expanded(trace['events']): + kind=event[0];counts[kind]+=1 + if kind=='add': + q=Q(event[3]);a,b=abs(q.numerator),q.denominator + need(a>0 and b&(b-1)==0,'dyadic ADD coefficient') + count=3+(a.bit_length()-1)+a.bit_count()+(b.bit_length()-1) + work+=count;macros+=1;real+=2*count + elif kind=='scale': + q=Q(event[2]);a,b=abs(q.numerator),q.denominator + need(a>0 and a&(a-1)==0 and b&(b-1)==0,'dyadic unit SCALE') + count=abs(a.bit_length()-b.bit_length())+int(q<0) + macros+=1;real+=2*count + elif kind in ('copy','clear','zero_buffer'): + macros+=1;real+=2 + # Explicit post-call exact clears make the private-zero return literal even + # for approximate recursive children. They preserve the ideal word endpoint. + postlude_clears=len(trace['zero_buffers'])+1 + real+=2*postlude_clears;macros+=postlude_clears + return dict(expanded_event_counts=dict(counts),scalar_macros=macros, + real_component_primitives=real,coefficient_work_primitives=work, + extra_exact_private_postlude_clears=postlude_clears, + conservative_macro_envelope=sum(counts.values())+postlude_clears) + +def main(): + parser=argparse.ArgumentParser();parser.add_argument('--variant',type=Path,required=True) + parser.add_argument('--owners',type=Path,required=True);parser.add_argument('--out',type=Path,required=True) + args=parser.parse_args();words={};local={};bindings={} + for orientation in ('forward','backward'): + p=args.variant/(orientation+'.json');words[orientation]=json.loads(p.read_bytes()) + local[orientation]=scalar_count(words[orientation]);local[orientation]['precision']=calculate(words[orientation]) + bindings[orientation+'_word_sha256']=hashlib.sha256(p.read_bytes()).hexdigest() + raw=args.owners.read_bytes();owners=json.loads(gzip.decompress(raw) if args.owners.suffix=='.gz' else raw) + groups=defaultdict(list) + for row in owners:groups[row['stage'],row['replica']].append(row) + helpers=len(words['forward']['helper_widths']);expected_helpers=set(map(int,words['forward']['helper_widths'])) + need(helpers==436,'selected fixed 44-alias variant') + need(set(groups)=={(stage,replica) for stage in range(9) for replica in range(8)},'all actual stage/replica calls') + for rows in groups.values(): + need(len(rows)==helpers and {r['helper'] for r in rows}==expected_helpers,'actual new physical helper roles') + calls=len(groups);replicas=8;bridges=9*replicas;m=73;v=112 + # Eight original private centre reserves safely cover the one COPY buffer; + # retain all 492 old local-helper reserves and add all 75 cleanup buffers. + live=896*replicas;retained_helper_reserve=492 + cleanup_buffers=max(len(w['zero_buffers']) for w in words.values()) + need(cleanup_buffers==75,'actual simultaneous FULL cleanup reserve') + centres,coefficient_work=8,2 + physical=live+retained_helper_reserve+cleanup_buffers+centres+coefficient_work + n=2*v+helpers+cleanup_buffers+1+coefficient_work + compiler=64*(m+1)**3 + children=sum(words['forward']['local_profile'].values()) + need(words['forward']['local_profile']==words['backward']['local_profile'],'same true paid profile') + macro_bound=max(x['conservative_macro_envelope'] for x in local.values()) + adapters=16*(macro_bound+children+n+centres)+64*m + primitive=max(x['real_component_primitives'] for x in local.values())+adapters*compiler + local_guard=max(x['precision']['local_sufficient_guard_bits'] for x in local.values()) + block_count=calls+bridges + summation_bits=(block_count-1).bit_length() + input_sup_to_l2_bits=(m+1)//2 + coordinate_defect_to_l2_bits=(m+1)//2 + optional_rounding_guard=local_guard+2*block_count+summation_bits+3+input_sup_to_l2_bits+coordinate_defect_to_l2_bits + ideal_local_bits=max(x['precision']['P_binary_upper'] for x in local.values()) + global_guard=ideal_local_bits+block_count+input_sup_to_l2_bits + payload=512 + # Endpoint-normalized data calls and bridge shears have stream-Hilbert + # operator norm <=2; helpers are content-independent unitary residuals. + # Thus P<=2^B, K<=B*2^B for B completed blocks. Applying the same feedback + # argument to the local guard gives the following deliberately coarse bill. + wrapper_count=2*helpers+2*v + wrapper_factors=calls*wrapper_count*2*(m-1) + bridge_additions=2*v*bridges + shifts=4*v*replicas;moves=2*shifts + outside_items=(6*calls*n+m*live+bridge_additions+shifts+moves+live+physical+wrapper_factors) + outside=8*outside_items+64*m + orientation_counts=Counter('forward' if stage%2==0 else 'backward' for stage,replica in groups) + scalar=sum(orientation_counts[o]*local[o]['real_component_primitives'] for o in local) + pervertex=scalar+bridge_additions+shifts+moves+live+(calls*adapters+outside)*compiler+64*physical + half=m//2;group=2**(half*half) + for j in range(1,half+1):group*=2**(2*j)-1 + stock=group*physical;logical=group*pervertex+64 + router=compiler*(logical+1)*(stock+1)**2 + # Source-independent mass overcharge: actual admitted profiles have positive + # deficit and therefore no more than m*live total paid rank. + streamed_mass=group*m*live + G,E,C0=2**30000,2**100000,2**210000 + literal=2*G*stock**2+8*streamed_mass+4*stock+4+32*m + total_mass=streamed_mass+E + row=2*stock.bit_length()+9909+252+2*payload + checks=dict(local_primitive_bound=primitive<2**48,logical=logical<2**6050, + router=router=1'), + exact_fine_grid_contract=dict(completed_child='C_r tensor powers are exact, with Gaussian dyadic denominators 2^(r*f) and coordinate row-sum at most 2^(r*f). No intermediate child rounding.', + active_depth_recurrence='A(e)<=A(rmax*floor(e/73))+s*floor(e/73)+E; for rmax<=72 and B=s+E>=8, A(e)<=2*B*e.', + outer_common_excess='(2*B+18)*d < 32*73*B^2*d; use Delta=ceil(C0*d) with C0=2^210000.', + literal_scope='Fresh literal charge includes every dyadic Horner primitive, private clear, monomial wrapper and route; native monomials introduce only unit Gaussian phases.'), + optional_rounding_robustness=dict(global_guard_coefficient=optional_rounding_guard, + scope='Separate sufficient feedback implication only, conditional on the explicit local Hilbert-defect contracts in precision receipts. It is not the inherited exact child contract and is not needed for the exact finite invoice.'), + wrapper_elementary_factors=wrapper_factors,actual_recounted_real_primitives=scalar, + conservative_outside_items=outside_items,conservative_pervertex=pervertex, + finite=dict(group_bits=group.bit_length(),physical_stock_bits=stock.bit_length(), + logical_bits=logical.bit_length(),router_bits=router.bit_length(),literal_bits=literal.bit_length(), + physical_row_coefficient=row,retained_row_coefficient=20161, + required_C0_bits=(32*m*total_mass*total_mass).bit_length(),C0_exponent=210000), + strict_checks=checks,source_bindings=bindings, + owner_table_sha256=hashlib.sha256(raw).hexdigest(), + inherited_conventions='m=73 cover order, chart compiler64*(m+1)^3, elementary route/family-selection bound, row9909+252, G=2^30000, E=2^100000, C0=2^210000', + scope='Fresh source-bound finite upper charges. All-size compiler, exact Gaussian-dyadic completed-child interface with active-depth common exponent, tape/route realization and analytic transfer remain inherited premises. Owner capacity/chronology/axes admission and exact moment/outer assembly are separate checked artifacts.') + args.out.write_text(json.dumps(output,indent=2)+'\n',encoding='utf-8') + print(output['status'],'physical',physical,'guard',global_guard,'row',row,'primitive_bits',primitive.bit_length()) + +if __name__=='__main__':main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/native.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/native.py new file mode 100644 index 0000000000..7478a0d237 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/native.py @@ -0,0 +1,176 @@ +"""Lower every actual operation of the complete credited borrowing trace. + +Native records are checked by a separately implemented signed-Pauli interpreter +and an exact Gaussian zero-column phase check. Checkpoints are immutable. +""" +import argparse +import gzip +import hashlib +import importlib.util +import json +import sys +import time +from functools import lru_cache +from pathlib import Path + +HERE = Path(__file__).resolve().parent +VENDOR = HERE / 'vendor' + +def load(name, path): + spec = importlib.util.spec_from_file_location(name, path) + module = importlib.util.module_from_spec(spec) + sys.modules[name] = module + spec.loader.exec_module(module) + return module + +CE = load('borrow_native_clifford', VENDOR / 'clifford_exact.py') +N = load('borrow_native_independent', VENDOR / 'native_verify.py') + +def need(condition, message): + if not condition: + raise ValueError(message) + +def canonical(vectors): + rows = {} + for value in vectors: + while value: + j = value.bit_length() - 1 + if j in rows: + value ^= rows[j] + else: + rows[j] = value + break + for j in sorted(rows): + for k in rows: + if k > j and rows[k] >> j & 1: + rows[k] ^= rows[j] + return tuple(rows[j] for j in sorted(rows, reverse=True)) + +@lru_cache(None) +def chart(vectors): + return CE.canonical_frame(9, canonical(vectors)) + +def inverse_columns(cols): + return CE.transpose(CE.inv(CE.transpose(cols, 9), 9), 9) + +def linear_columns(cols, value): + out = 0 + for i, c in enumerate(cols): + if value >> i & 1: + out ^= c + return out + +def change_columns(old, new): + a, b = CE.basis_extend(old, 9), CE.basis_extend(new, 9) + return tuple(linear_columns(b, col) for col in inverse_columns(a)) + +def logical_operator(warp): + cols = change_columns(tuple(warp['from_basis']), tuple(warp['to_basis'])) + inverse = inverse_columns(cols) + transposed_inverse = CE.transpose(inverse, 9) + quadratic = tuple((1 << i) ^ linear_columns(transposed_inverse, inverse[i]) for i in range(9)) + return CE.perm(9, CE.transpose(cols, 9)).then(CE.diag(9, quadratic)) + +def make_jobs(trace, literal_bound=False): + logical = {w['operator']: logical_operator(w) for w in trace['warps']} + fourier = CE.fourier(9) + dual = {symbol: fourier.adjoint().then(op.adjoint()).then(fourier) + for symbol, op in logical.items()} + jobs, event_refs = {}, [] + def add(kind, metadata, actual, rank): + payload = [kind, metadata] + if literal_bound: + payload.append(actual.circuit) + key = hashlib.sha256(json.dumps(payload, sort_keys=True, separators=(',', ':')).encode()).hexdigest() + jobs.setdefault(key, dict(kind=kind, metadata=metadata, actual=actual, rank=rank)) + return key + for index, event in enumerate(trace['events']): + kind = event[0] + if kind == 'transport': + old, new = canonical(event[2]), canonical(event[3]) + key = add(kind, dict(old=old, new=new), chart(old).adjoint().then(chart(new)), event[4]) + elif kind in ('handoff', 'dual_handoff'): + old, new, symbol = canonical(event[2]), canonical(event[3]), event[4] + reference = dual if kind == 'dual_handoff' else logical + op = chart(old).adjoint().then(reference[symbol]).then(chart(new)) + key = add(kind, dict(old=old, new=new, symbol=symbol), op, 0) + elif kind in ('logical_monomial', 'dual_monomial'): + symbol, frame = event[2], canonical(event[3]) + reference = dual if kind == 'dual_monomial' else logical + op = reference[abs(symbol)] + if symbol < 0: + op = op.adjoint() + op = chart(frame).adjoint().then(op).then(chart(frame)) + key = add(kind, dict(frame=frame, symbol=symbol), op, 0) + else: + key = None + event_refs.append(key) + residual_refs = {} + for register, endpoints in trace['endpoints'].items(): + if int(register) < 224: + continue + old, new = canonical(endpoints['initial']), canonical(endpoints['final']) + residual_refs[register] = add('helper_residual', dict(old=old, new=new), + chart(old).adjoint().then(chart(new)), len(new) - len(old)) + return jobs, event_refs, residual_refs + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument('--limit', type=int) + parser.add_argument('--orientation', choices=('forward', 'backward'), default='forward') + parser.add_argument('--trace', type=Path) + parser.add_argument('--out-dir', type=Path) + args = parser.parse_args() + source = args.trace or HERE / (args.orientation + '-physical-events.json') + trace = json.loads(source.read_bytes()) + jobs, refs, residuals = make_jobs(trace, literal_bound=bool(args.trace)) + destination = args.out_dir or HERE / ('native-' + args.orientation) + records_dir = destination / 'records' + records_dir.mkdir(parents=True, exist_ok=True) + started = time.monotonic() + count = 0 + retained = {} + def operation_key(circuit, rank): + return hashlib.sha256(json.dumps([circuit,rank],separators=(',',':')).encode()).hexdigest() + for orientation in ('forward','backward'): + for path in (HERE/('native-'+orientation)/'records').glob('*.json'): + record=json.loads(path.read_bytes()) + retained.setdefault(operation_key(record['literal_expected'],record['recipe']['rank']),record) + for key, job in sorted(jobs.items()): + path = records_dir / (key + '.json') + if path.exists(): + continue + previous=retained.get(operation_key(job['actual'].circuit,job['rank'])) + if previous is not None: + recipe=previous['recipe'] + else: + recipe, compiled = CE.lower_child(job['actual']) + need(recipe['rank'] == job['rank'], ('actual physical rank', job['kind'], job['metadata'], recipe['rank'], job['rank'])) + N.equal(N.recipe(recipe), tuple(job['actual'].circuit), 'complete borrowing ' + job['kind']) + record = dict(kind=job['kind'], metadata=job['metadata'], recipe=recipe, + literal_expected=job['actual'].circuit, + signed_pauli_rows=job['actual'].tab, + exact_gaussian_zero_state=(previous.get('exact_gaussian_zero_state') if previous is not None else + CE.monomial(compiled)[1] if recipe['rank'] == 0 else None)) + path.write_text(json.dumps(record, separators=(',', ':')) + '\n', encoding='utf-8') + count += 1 + if count % 50 == 0: + print('NATIVE_COMPILED', count, 'new;', len(list(records_dir.glob('*.json'))), '/', len(jobs), 'seconds', round(time.monotonic() - started, 2), flush=True) + if args.limit and count >= args.limit: + break + present = {p.stem for p in records_dir.glob('*.json')} + complete = set(jobs) <= present + report = dict(status='PASS_ALL_' + args.orientation.upper() + '_NATIVE_RECIPES' if complete else 'PARTIAL_' + args.orientation.upper() + '_NATIVE_CHECKPOINT', + jobs=len(jobs), present=len(present), newly_compiled=count, + trace_sha256=hashlib.sha256(source.read_bytes()).hexdigest(), + compiler_sha256=hashlib.sha256((VENDOR / 'clifford_exact.py').read_bytes()).hexdigest(), + independent_checker_sha256=hashlib.sha256((VENDOR / 'native_verify.py').read_bytes()).hexdigest(), + events=len(refs), helper_residuals=len(residuals), + literal_bound_keys=bool(args.trace), + limitation='Finite exact native operators for the ' + args.orientation + ' word only; whole-call hosting, precision and all-size admission separate.') + (destination / 'STATUS.json').write_text(json.dumps(report, indent=2) + '\n', encoding='utf-8') + if complete: + out = dict(event_recipe_refs=refs, helper_residual_refs=residuals, + records={key: json.loads((records_dir / (key + '.json')).read_bytes()) for key in sorted(jobs)}) + (destination / ('NATIVE-' + args.orientation.upper() + '.json.gz')).write_bytes(gzip.compress((json.dumps(out, separators=(',', ':')) + '\n').encode(), mtime=0)) + print(json.dumps(report, indent=2), flush=True) diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/precision.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/precision.py new file mode 100644 index 0000000000..da5e30fc80 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/precision.py @@ -0,0 +1,106 @@ +"""All-cut noncancellation Hilbert-norm envelope for literal local words. + +This establishes a sufficient implication under an explicit completed-instruction +defect contract. It does not instantiate rounded recursive children or an all-size +stock/row envelope. Native Clifford factors and monomial adapters are unitary. +""" +import argparse +import hashlib +import json +from fractions import Fraction as Q +from pathlib import Path + +def need(ok,label): + if not ok:raise ValueError(label) + +def binary_ceiling(x): + need(x>0,'positive norm bound') + power=max(0,x.numerator.bit_length()-x.denominator.bit_length()) + while Q(2)**power1 and type(e[1]) is int:used.add(e[1]) + work=max(used)+1 + yield ['zero_buffer',work,[],trace['active_registers'][0]] + for e in expanded(trace['events']): + kind=e[0] + if kind=='add': + _,target,source,q,F=e;q=Q(q);a,b=abs(q.numerator),q.denominator + need(a>0 and b&(b-1)==0,'dyadic Horner coefficient') + yield ['clear',work,F] + for index,bit in enumerate(bin(a)[2:]): + if index:yield ['scale',work,'2',F] + if bit=='1':yield ['add',work,source,'1',F] + for _ in range(b.bit_length()-1):yield ['scale',work,'1/2',F] + yield ['add',target,work,str(1 if q>0 else -1),F] + yield ['clear',work,F] + elif kind=='scale': + _,r,q,F=e;q=Q(q);a,b=abs(q.numerator),q.denominator + need(a>0 and a&(a-1)==0 and b&(b-1)==0,'power-of-two scalar unit') + if q<0:yield ['scale',r,'-1',F] + power=a.bit_length()-b.bit_length() + for _ in range(abs(power)):yield ['scale',r,'2' if power>0 else '1/2',F] + else:yield e + # The explicit implementation postlude clears every owned array exactly. + private={e[1] for e in trace['events'] if e[0] in ('copy','zero_buffer')} + for r in sorted(private):yield ['clear',r,[]] + yield ['clear',work,[]] + +def calculate(trace): + ideal={r:Q(1) for r in trace['active_registers']} + errors={r:Q(0) for r in trace['active_registers']} + P,K=Q(1),Q(0);peakP=peakK=None;count=0 + for index,e in enumerate(primitive_expansion(trace)): + count+=1;kind=e[0];r=e[1] + if kind=='add': + q=abs(Q(e[3]));s=e[2] + ideal[r]+=q*ideal[s];errors[r]+=q*errors[s];local=2*(1+q) + elif kind=='scale': + q=abs(Q(e[2]));ideal[r]*=q;errors[r]*=q;local=2*max(Q(1),q) + elif kind=='copy': + s=e[2];ideal[r]=ideal[s];errors[r]=errors[s];local=Q(1) + elif kind in ('clear','zero_buffer'): + # Writing the exact zero bit-pattern incurs no numerical rounding. + ideal[r]=errors[r]=Q(0);local=Q(0) + elif kind in ('transport','handoff','logical_monomial','dual_handoff','dual_monomial','zero_reframe'): + # Physical native factors are unitary; zero reframe is literal identity. + need(r in ideal,'allocated physical stream') + local=Q(1) + else:raise ValueError(('unsupported physical instruction',kind)) + errors[r]+=local + if ideal[r]>P:P=ideal[r];peakP=[index,r] + if errors[r]>K:K=errors[r];peakK=[index,r] + p,k=binary_ceiling(P),binary_ceiling(K) + guard=p+k+3 + u=Q(1,2**guard) + need(K*u=2*v} + trace=dict(status='EXPORTED_COMPLETE_FORWARD_PHYSICAL_EVENTS_NATIVE_LOWERING_PENDING', + source_replay_sha256='0f9a47cdaeed0e22938c63251c232d6f23aeee2b62b0b1223cd4473c5428bcf7',events=native,endpoints=endpoints, + helper_widths=helpers,zero_buffers=buffer_map,copy_register=copy_register, + original_register_count=n,active_registers=active,warps=warps, + local_profile=dict(sorted(profile.items())),scalar_counts=dict(Counter(e[0] for e in native))) + (a.output.parent/'forward-physical-events.json').write_text(json.dumps(trace,indent=2)+'\n') + a.output.write_text(json.dumps(out,indent=2)+'\n');print('PASS_FIXED_BORROWED_WORD columns='+str(len(active))+' removed='+str(len(removed))+' cold='+str(len(restored))) + if a.framed_word: + data={'start':{str(r):w['frames'][w['start'][r]] for r in active}, + 'final':{str(r):frame_basis(cur[r]) for r in active},'steps':framed_steps, + 'copy_profile':dict(Counter(len(w['frames'][f]) for f in w['center_frames'])), + 'handoff_profile':dict(Counter(x.get('physical_handoff_rank',0) for x in warps if x.get('physical_handoff_rank',0))), + 'original_profile':dict(sorted(profile.items())),'h':w['h']} + a.framed_word.write_text(json.dumps(data,separators=(',',':'))+'\n') + + +if __name__=='__main__':main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/FRAME-SA504-NOTICE.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/FRAME-SA504-NOTICE.md new file mode 100644 index 0000000000..f7a8f861be --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/FRAME-SA504-NOTICE.md @@ -0,0 +1,24 @@ +# Attribution and derivative scope + +Prepared for icekylinx with OpenAI GPT-6 Astra assistance; Apache-2.0. +Copyright 2026 icekylinx. Local review and submission packaging used OpenAI Codex. +The new work is the same-partition 504-helper scalar architecture, its frozen +entrance selection, delivery-frame coalescing, literal global allocation, +targeted certificate reuse and complete new finite recount. + +The included Hamming inputs, Gaussian-dyadic Clifford compiler, primitive +scalar routines, rational moment engines and outer assembly descend from the +authors recorded in `SOURCES.json` and `inputs/PR523-ANCESTRY.json`. Original +source notices are retained. In particular, DreamingOfClouds' gauge/kernel +and paid-wrapper files retain their notice crediting Claude Opus 5.5 assistance; +those files are not presented as newly authored work. + +Dugongue's PR522 rank-sector coordinate-total principle was inspected and is +credited as a related idea. This increment does not import its ranksector99 +patch or add its reported gain arithmetically. Higherbeing's PR520 face-packet +geometry helped direct the structural investigation; its code is not copied. +The latest numerical comparison, Rohan Arun's PR527, is a comparison source. + +The reproduction wrapper deliberately separates fresh changed-word/global +checks from reuse of the byte-pinned completed baseline and unchanged bit +operator. Old finite receipts never stand in for the changed local word. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR1862-NOTICE.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR1862-NOTICE.md new file mode 100644 index 0000000000..24461eddc1 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR1862-NOTICE.md @@ -0,0 +1,50 @@ +# Source attribution and license + +New research, code, and verification used substantial OpenAI Codex +assistance under the user's direction. Separate agents developed the +input construction and reviewed its algebra, native phases, and literal +programs. No independent human review is claimed. New material uses the +repository's Apache-2.0 license. + +The package reads immutable source objects at commit +`bce54fb3f895cbe5cd4c0eb477c3afeed100190f`. It does not change the frozen +source archive or copy a predecessor package. Exact source pins are in +[SOURCES.json](SOURCES.json). The original notices are retained as exact +files under `notices/`. + +The local word comes from the gauge44 lineage. Its +[notice](https://github.com/CrocSwap/integer-mult-bounds/blob/bce54fb3f895cbe5cd4c0eb477c3afeed100190f/research/gauge44-slots-t16/NOTICE.md) +credits @icekylinx, with OpenAI GPT-6 Astra assistance, for the source +gauge and related local construction. The +[construction notice](https://github.com/CrocSwap/integer-mult-bounds/blob/bce54fb3f895cbe5cd4c0eb477c3afeed100190f/research/gauge44-slots-t16/construction/NOTICE.md) +preserves the factory, K-triangle, porting, joint-prefix, and HPE lineage. + +The retained +[t16 notice](https://github.com/CrocSwap/integer-mult-bounds/blob/bce54fb3f895cbe5cd4c0eb477c3afeed100190f/research/gauge44-slots-t16/t16/NOTICE.md) +credits Hao Yan (@FanPUYen) for the integration; Jeremy Kothe +(@jeremykothe2) for the [PR #799](https://github.com/CrocSwap/integer-mult-bounds/pull/799) +local supplier; and Chafik Boukhalfa (@chafreaky) for the +[PR #741](https://github.com/CrocSwap/integer-mult-bounds/pull/741) +frame work. These sources and their earlier attribution are retained +inputs, not new results of this package. Name-to-account evidence is in +the pinned +[contributor record](https://github.com/CrocSwap/integer-mult-bounds/blob/bce54fb3f895cbe5cd4c0eb477c3afeed100190f/CONTRIBUTORS.md). + +The complete +[upstream notice](https://github.com/CrocSwap/integer-mult-bounds/blob/bce54fb3f895cbe5cd4c0eb477c3afeed100190f/research/gauge44-slots-t16/UPSTREAM-NOTICE.md) +retains DreamingOfClouds, earlier cold-end and joint-prefix work, other +contributors, source pins, licenses, and AI disclosures. It is copied +without edits. The selected slot-word lineage also credits @rohanarun +for [PR #987](https://github.com/CrocSwap/integer-mult-bounds/pull/987) +and NewBArch for [PR #869](https://github.com/CrocSwap/integer-mult-bounds/pull/869) +in the pinned +[local README](https://github.com/CrocSwap/integer-mult-bounds/blob/bce54fb3f895cbe5cd4c0eb477c3afeed100190f/research/gauge44-slots/README.md). +NewBArch is the credited source name; no account identity is inferred. + +The portable source loader, sparse replay, and exact native interpreter +come from this centre-encoding campaign. They retain the same source +objects used in [PR #1719](https://github.com/CrocSwap/integer-mult-bounds/pull/1719), +but this package does not depend on that PR being merged. The new verifier +does not import the source-code producer or the exploratory native +compiler. The historical global owner map and final invoice are not +replayed or improved by this local package. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR1972-NOTICE.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR1972-NOTICE.md new file mode 100644 index 0000000000..eb4ac7c07f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR1972-NOTICE.md @@ -0,0 +1,11 @@ +New material: Rohan Arun, with substantial OpenAI Codex assistance. Apache-2.0. +Vendored PR1940 verifier and schedule: Adam Gall, with substantial OpenAI +Codex (gpt-6-astra) assistance; Apache-2.0. The source docstring is retained. +The P104 fixture comes from Josie Elliston / PR1071, with substantial Claude +Opus 5.5 assistance, through the hash-pinned PR1940 input. The decoder, +32-path bound and their lineage belong to PR1071, Ali Romman / PR1125, +themilkman216 / PR968, DreamingOfClouds / PR717, PR446/PR595 and GamingPuzzled +/ Joel Pulikkan / PR620. None of those results is claimed new here. +The generic dirty-cancellation identity is not claimed new. PR673 and the +existing dirty-helper source/target compensation methods are adjacent +precedents. No literature-priority guarantee or independent human review. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR1991-README.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR1991-README.md new file mode 100644 index 0000000000..f187d1ac0b --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR1991-README.md @@ -0,0 +1,91 @@ +# Native P104 low-rank channel + +Concrete finite Clifford recipes for the complete low-rank channel in +[Rohan Arun's PR1972](https://github.com/CrocSwap/integer-mult-bounds/pull/1972), +head `fd82d172f46933fd253c2c6de4bd5eaff5429cfd`. Its scalar/frame construction +is retained, not claimed new. This package replaces its conditional free-chart +realization with exact Gaussian-dyadic primitive recipes and intrinsic data +endpoint adapters. + +**No complete P104 supplier or new kappa.** The `I-G/2` side correction, +chronological global hosting, precision and all-size admission remain open. +The selected frontier is unchanged. + +## Result + +For the frozen 104 root ports, source logical values `x`, target values `y` +and 72 independent arbitrary dirty helpers, the channel implements +`y <- y + (G/2)x`, with the prescribed source/target transports. All 72 helper +logical values are restored at the full frame. A separate owned private copy +has eight explicit allocate/copy/descend/discard lifetimes under the inherited +private-copy contract; it is not one of the 72 dirty inputs. + +[EXPECTED.json](EXPECTED.json) records: + +- 740 distinct native transport recipes covering all 1410 paid occurrences; +- 208 monomial data endpoint adapters: 104 source entries and 104 target exits; +- all 280 arbitrary register input columns in a complete framed induction; +- 14,400 physical root-pairing checks and 4096 exact Gaussian private-drop columns; +- five rejecting controls, including global phase and missing query compensation/cleanup. + +The exact profile remains +`1074 H1 + 58 H2 + 86 H3 + 104 H4 + 48 H5 + 24 H6 + 8 H7 + 8 H8`. +Its rank mass is **2368 = 2312 boundary coverage + COPY56**. This is the same +local boundary invoice as PR1972, not a globally hosted recursive price. + +The compressed root labels require a nontrivial Gram isometry before using +ordinary physical binary dot products. [PROOF.md](PROOF.md) gives that binding, +the native operator check and the full-word induction. + +## Reproduce + +Standard-library Python. Default replay is read-only, needs no network or Git, +and imports neither the channel producer nor the Clifford compiler: + +```sh +python3 -I -O -B research/p104-native-channel/verify.py +``` + +Also rebuild every recipe in temporary storage and compare the complete +decompressed payload: + +```sh +python3 -I -O -B research/p104-native-channel/verify.py --regenerate +``` + +The registered hook runs both checks with a 300-second limit: + +```sh +python3 scripts/research_package.py --package research/p104-native-channel --check-only +python3 scripts/research_package.py --package research/p104-native-channel +``` + +The generic metadata check needs Git provenance. The direct command also works +from a read-only source export. `--out PATH` writes a receipt exclusively to a +new caller-selected file. Source hashes and exact origins are in +[SOURCES.json](SOURCES.json); [TASK.json](TASK.json) records the changed audit scope. +Compression bytes are not used to compare regenerated mathematics. + +## Open next questions + +Integrate a fully charged `I-G/2` side word while the required source/target +frames remain available. Appending it after source completion is not justified. +Then assign actual chronological owners to the completed word, count all idle +and residual children, and compare the exact recursive moment. Those steps, +plus precision and all-size lifting, are required before a supplier or kappa +claim. Concrete first steps and blockers are in `TASK.json`. + +## Credit + +Rohan Arun / PR1972 supplies the complete scalar/frame channel; Adam Gall / +PR1940 supplies the schedule. The P104 fixture and decoder derive from Josie +Elliston / PR1071, with Ali Romman / PR1125 and the PR968/717/446/595/620 lineage. +The exact compiler is retained through icekylinx's frame-sa504 package; the +independent Gaussian interpreter comes from Ali Romman's PR1862, also retained +in PR1931. Exact predecessor notices are under `notices/`. + +Native lowering, the independent physical binding and framed induction were +prepared for Chafik Boukhalfa with substantial OpenAI Codex assistance. +Apache-2.0. No independent human review, proof assistant or literature-priority +guarantee is claimed. The fresh overlap screen covered 470 open PRs and 614 +discussions; its limits are recorded in `SOURCES.json`. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR2226-NOTICE.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR2226-NOTICE.md new file mode 100644 index 0000000000..5ea70357a0 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR2226-NOTICE.md @@ -0,0 +1,22 @@ +# Attribution and license + +New construction, proof text and checkers: Cinna Davis (@1337isnot1337), with +substantial OpenAI Codex assistance, Apache-2.0. + +The selected complex canonical word is the PR1042 ancestry retained in PR1094. +Jeremy Kothe, Rohan Arun, Hao Yan, Chafik Boukhalfa, icekylinx, Jonathan Capalbo +and the source package's full provider/frame/controller lineage retain credit. +Ali Romman supplied the source-bound fixed-block theorem and certificate; +Thomas DiFiore supplied the earlier block discovery. The borrowing construction +does not attribute its changed interface to those predecessor results. + +`price_engine.py` is an unchanged copy of the incorporated +`research/complex-local-word-ceiling/ceiling.py`: Douglas Colkitt, with Claude +(Anthropic) assistance; original SPDX/license context and docstring retained. +General Clifford frame and exact Gaussian normal forms are inherited from +`notes/general-clifford-frames.tex` and the repository's credited phase/compiler +lineage. The package independently checks its actual local Gaussian products. + +Source hashes, base commits and the copied engine's Git blob are in SOURCES.json. +No full native supplier, all-size compiler, precision theorem or new kappa is +claimed. No independent human review is claimed. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR2226-replay.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR2226-replay.py new file mode 100644 index 0000000000..1bbe590206 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/vendor/PR2226-replay.py @@ -0,0 +1,293 @@ +#!/usr/bin/env python3 +"""Exact replay over a free operator algebra for nominated dirty borrowing. + +The current implementation closes cleanup at FULL on every non-target field. +It prices that choice explicitly. This is not a hosted supplier admission. +""" +import argparse,gzip,json,time +from collections import Counter +from fractions import Fraction as Q +from pathlib import Path + + +def span(basis): + s={0} + for x in basis:s|={t^x for t in s.copy()} + return frozenset(s) + + +def incident(e):return e[1:3] if e[0]=='add' else e[1:2] + + +def add(dst,src,c): + for key,value in list(src.items()): + z=dst.get(key,Q(0))+c*value + if z:dst[key]=z + elif key in dst:del dst[key] + + +def warp(row,op): + out={} + for (var,word),value in row.items(): + word=word[1:] if word and word[0]==-op else (op,)+word + out[var,word]=value + return out + + +def main(): + p=argparse.ArgumentParser();p.add_argument('--word',type=Path,required=True) + p.add_argument('--blocks',type=Path);p.add_argument('--inventory',type=Path,required=True) + p.add_argument('--recover-cold',action='store_true') + p.add_argument('--framed-word',type=Path) + p.add_argument('--output',type=Path,required=True);a=p.parse_args();start=time.monotonic() + w=json.loads(gzip.decompress(a.word.read_bytes()));blocks=json.loads(a.blocks.read_text())['sources'][0]['blocks'] if a.blocks else [] + inv=json.loads(a.inventory.read_text());pairs=inv['pairs'] + assert len(pairs)==80 and sum(x['low'] for x in pairs)==32 + assert all(x.get('handoff_paid_rank',0)==0 for x in pairs) + v=w['v'];n=2*v+w['R'];full=span([1<>i)&1)<>j&1);a[j],a[k]=a[k],a[j] + for i in range(n): + if i!=j and a[i]>>j&1:a[i]^=a[j] + assert [r&((1<>n for r in a) +def basis_extend(cols,n): + out=[] + for x in list(cols)+[1<len(out):out.append(x) + assert len(out)==n + return tuple(out) +def nullspace(rows,n): + a=list(rows);p=[];k=0 + for j in range(n): + t=next((i for i in range(k,len(a)) if a[i]>>j&1),None) + if t is None:continue + a[t],a[k]=a[k],a[t] + for i in range(len(a)): + if i!=k and a[i]>>j&1:a[i]^=a[k] + p.append(j);k+=1 + return tuple((1<>j)&1)<>i&1:out=pmul(out,T[i]) + for i in range(n): + if z>>i&1:out=pmul(out,T[n+i]) + return out +def compose(A,B):return tuple(apply(A,p) for p in B) +def inverse(T): + n=len(T)//2;cols=[x|(z<>n;u,v,p=apply(T,(x,z,0));assert (u,v)==identity(n)[j][:2];out.append((x,z,(-p)%4)) + return tuple(out) +def qvalue(Q,alpha,x,n): + return (sum(alpha[i] for i in range(n) if x>>i&1)+2*sum(parity((Q[i]>>(i+1))&(x>>(i+1))) for i in range(n) if x>>i&1))%4 +def mul_i(a,p): + x,y=a;p%=4 + return [(x,y),(-y,x),(-x,-y),(y,-x)][p] +@dataclass +class Op: + n:int + tab:tuple + circuit:list + def then(self,b): + assert self.n==b.n + return Op(self.n,compose(b.tab,self.tab),self.circuit+b.circuit) + def adjoint(self): + circ=[] + for e in reversed(self.circuit): + if e[0]=='P':circ.append(('P',inv(e[1],self.n))) + elif e[0]=='D':circ.append(('D',e[1],tuple((-x)%4 for x in e[2]))) + elif e[0]=='C':circ.append(('C',e[1],-e[2])) + elif e[0]=='phase':circ.append(('phase',(-e[1])%4)) + else:circ.append(e) + return Op(self.n,inverse(self.tab),circ) + def zero_state(self,basis=0): + N=1<>i)&1 for i in range(n)) + assert all(alpha[i]%2==(Q[i]>>i)&1 for i in range(n)) + return Op(n,tuple([(1<>i&1 else 0,1<>i&1 else 0) for i in range(n)]),[('C',q,sgn)]) +def xgate(n,b):return Op(n,tuple([(1<>i)&1)) for i in range(n)]),[('X',b)]) +def fourier(n,axes=None): + out=ident(n) + for i in range(n) if axes is None else axes:out=out.then(cgate(n,1<len(selected):picks.append(j);selected.append(x) + assert len(picks)==r and all(x>>r==0 for x in Bcols) + Br=transpose(selected,r);Bi=inv(Br,r);Dr=transpose([Dcols[j]&((1<> j) & 1) << i for i, row in enumerate(matrix)) for j in range(len(matrix))) + + +def inverse(matrix): + n = len(matrix) + rows = [row | (1 << (n+i)) for i, row in enumerate(matrix)] + for j in range(n): + pivot = next((i for i in range(j, n) if rows[i] >> j & 1), None) + require(pivot is not None, 'singular binary chart') + rows[j], rows[pivot] = rows[pivot], rows[j] + for i in range(n): + if i != j and rows[i] >> j & 1: + rows[i] ^= rows[j] + require([row & ((1 << n)-1) for row in rows] == [1 << i for i in range(n)], 'binary inverse') + return tuple(row >> n for row in rows) + + +def extend(vectors, n): + columns, pivots = [], {} + for value in tuple(vectors) + tuple(1 << i for i in range(n)): + reduced = value + while reduced: + j = reduced.bit_length()-1 + if j not in pivots: + pivots[j] = reduced + columns.append(value) + break + reduced ^= pivots[j] + require(len(columns) == n, 'frame basis extension') + return tuple(columns) + + +def adjoint(circuit): + out = [] + for gate in reversed(circuit): + if gate[0] == 'P': + out.append(('P', inverse(gate[1]))) + elif gate[0] == 'D': + out.append(('D', gate[1], tuple(-a % 4 for a in gate[2]))) + elif gate[0] == 'C': + out.append(('C', gate[1], -gate[2])) + elif gate[0] == 'phase': + out.append(('phase', -gate[1] % 4)) + else: + require(gate[0] == 'X', 'adjoint gate type') + out.append(gate) + return tuple(out) + + +def fourier(rank): + return tuple(('C', 1 << i, 1) for i in range(rank)) + + +def frame(vectors, n): + rank = len(vectors) + if rank == 0: + return () + if rank == n: + return fourier(n) + matrix = transpose(extend(vectors, n)) + columns = transpose(inverse(matrix)) + quadratic = tuple((1 << i) ^ sum(parity(a & b) << j for j, b in enumerate(columns)) for i, a in enumerate(columns)) + diagonal = tuple((row >> i) & 1 for i, row in enumerate(quadratic)) + chart = (('P', matrix), ('D', quadratic, diagonal)) + return adjoint(chart) + fourier(rank) + chart + + +def monomial(record): + matrix = tuple(record['matrix_rows']) + quadratic = tuple(record['quadratic_rows']) + linear = tuple(record['linear_Z4']) + require(transpose(quadratic) == quadratic, 'quadratic symmetry') + require(len(matrix) == len(quadratic) == len(linear) == 9, 'monomial dimension') + require(all(a % 2 == (quadratic[i] >> i) & 1 for i, a in enumerate(linear)), 'diagonal parity') + inverse(matrix) + return (('D', quadratic, linear), ('P', matrix), ('X', record['offset']), ('phase', record['phase'])) + + +def recipe(record): + rank = record['rank'] + require(isinstance(rank, int) and 0 <= rank <= 9, 'child rank') + if rank == 0: + require(record['right'] is None, 'rank zero right factor') + return monomial(record['left']) + return monomial(record['right']) + fourier(rank) + monomial(record['left']) + + +def tableau(circuit, n=9): + rows = [(1 << i, 0, 0) for i in range(n)] + [(0, 1 << i, 0) for i in range(n)] + for gate in circuit: + tag = gate[0] + out = [] + inverse_transpose = transpose(inverse(gate[1])) if tag == 'P' else None + for x, z, phase in rows: + if tag == 'P': + x, z = mul(gate[1], x), mul(inverse_transpose, z) + elif tag == 'X': + phase += 2 * parity(z & gate[1]) + elif tag == 'C': + if parity(z & gate[1]): + x ^= gate[1] + phase += 3 * gate[2] + elif tag == 'D': + quadratic, linear = gate[1:] + phase += sum(linear[i] for i in range(n) if x >> i & 1) + phase += 2 * sum(parity((quadratic[i] >> (i+1)) & (x >> (i+1))) for i in range(n) if x >> i & 1) + z ^= mul(quadratic, x) + else: + require(tag == 'phase', 'tableau gate type') + out.append((x, z, phase % 4)) + rows = out + return tuple(rows) + + +@lru_cache(maxsize=512) +def action(gate): + if gate[0] == 'P': + return tuple(mul(gate[1], x) for x in range(512)) + require(gate[0] == 'D', 'state action type') + quadratic, linear = gate[1:] + return tuple((sum(linear[i] for i in range(9) if x >> i & 1) + + 2 * sum(parity((quadratic[i] >> (i+1)) & (x >> (i+1))) for i in range(9) if x >> i & 1)) % 4 for x in range(512)) + + +def rotate(pair, power): + a, b = pair + return ((a, b), (-b, a), (-a, -b), (b, -a))[power % 4] + + +def state(circuit, basis=0): + value, denominator = {basis: (1, 0)}, 0 + for gate in circuit: + tag = gate[0] + result = {} + if tag == "C": + direction, sign = gate[1:] + for address, (a, b) in value.items(): + for target, pair in ((address, (a - sign*b, b + sign*a)), + (address ^ direction, (a + sign*b, b - sign*a))): + old = result.get(target, (0, 0)) + result[target] = (old[0] + pair[0], old[1] + pair[1]) + denominator += 1 + else: + for address, pair in value.items(): + if tag == "P": + target = mul(gate[1], address) + elif tag == "X": + target = address ^ gate[1] + else: + target = address + if tag == "D": + pair = rotate(pair, action(gate)[address]) + else: + require(tag == "phase", "Unknown Gaussian state gate") + pair = rotate(pair, gate[1]) + require(target not in result, "A monomial address map is not injective") + result[target] = pair + value = {address: pair for address, pair in result.items() if pair != (0, 0)} + while denominator and all((a | b) % 2 == 0 for a, b in value.values()): + denominator -= 1 + value = {address: (a//2, b//2) for address, (a, b) in value.items()} + return value, denominator + + +def equal(actual, expected, label): + require(tableau(actual) == tableau(expected), label + ": signed Pauli action") + require(state(actual) == state(expected), label + ": exact global phase") + + +def verify(data, certificate): + word = data["word"] + original, clean = data["programs"], data["clean_programs"] + source_recipes = data["recipes"] + residuals = data["residuals"]["recipes"] + native_tags = {"climb", "adapter", "residual_adapter"} + for orientation in ("forward", "backward"): + native = [event for event in clean[orientation] if event[0] in native_tags] + require(native == [event for event in original[orientation] if event[0] in native_tags], + "A retained native event changed") + require(len(native) == 5056, "The retained native event count changed") + frames = [frame(tuple(vectors), 9) for vectors in word["frames"]] + used = {(event[3], event[4]) for program in clean.values() + for event in program if event[0] == "climb"} + children = {(record["old"], record["new"]): record for record in source_recipes["children"]} + require(len(children) == len(source_recipes["children"]) and used <= set(children), + "The retained child recipe coverage is incomplete") + for old, new in sorted(used): + record = children[old, new] + require(record["rank"] == len(word["frames"][new]) - len(word["frames"][old]), + "A retained child has the wrong rank") + equal(recipe(record), adjoint(frames[old]) + frames[new], "Retained child") + endpoints = {(record["role"], record["port"]): record + for record in source_recipes["endpoint_adapters"]} + require(len(endpoints) == len(source_recipes["endpoint_adapters"]) + and set(endpoints) == {(role, p) for role in ("source", "target") for p in range(112)}, + "The endpoint adapter coverage is incomplete") + for (role, port), record in endpoints.items(): + direction = word["ports"][port] + conventional = (("C", direction, 1),) if role == "source" else ( + ("C", direction, -1),) + fourier(9) + index = word["start"][port] if role == "source" else word["final"][112 + port] + require(record["frame"] == index and record["rank"] == 0, "An endpoint frame changed") + equal(recipe(record), adjoint(conventional) + frames[index], "Endpoint adapter") + used_residuals = set() + for program in clean.values(): + for event in program: + if event[0] != "residual_adapter": + continue + _, position, register, entry, *tail = event + terminal = tail[0] if tail else 1 + key = str(entry) if terminal == 1 else str(entry) + ":" + str(terminal) + require(key in residuals and position in ("pre", "post"), "A residual binding changed") + require(entry == word["start"][register] and terminal == word["final"][register], + "A residual endpoint changed") + used_residuals.add(key) + for key in sorted(used_residuals): + record = residuals[key] + equal(recipe(record), adjoint(frames[record["frame"]]) + frames[record.get("terminal", 1)], + "Residual adapter") + records = certificate["input_recipes"] + ports = certificate["compensation_ports"] + require([record["port"] for record in records] == ports and len(records) == 8, + "The eight input recipes changed") + columns = 0 + for item in records: + port, label = item["port"], item["label"] + require(label == word["ports"][port] and item["canonical_start_frame_id"] == word["start"][port], + "An input recipe has the wrong supplier label") + record = item["recipe"] + require(record["rank"] == 1, "An input copy has the wrong child rank") + for side in ("left", "right"): + factor = record[side] + require(factor["quadratic_rows"] == [0] * 9 and factor["linear_Z4"] == [0] * 9 + and factor["offset"] == factor["phase"] == 0, + "An input factor is not an address permutation") + actual, expected = recipe(record), (("C", label, 1),) + equal(actual, expected, "Positive inherited input line") + for basis in range(512): + require(state(actual, basis) == state(expected, basis), "A complete input Gaussian column differs") + columns += 1 + negative = monomial(record["right"]) + (("C", 1, -1),) + monomial(record["left"]) + require(tableau(negative) != tableau(expected) or state(negative) != state(expected), + "An inverse input child was not detected") + phased = actual + (("phase", 1),) + require(tableau(phased) == tableau(actual) and state(phased) != state(expected), + "An input phase-only error was not detected") + return {"status": "PASS_EXACT_INPUT_NATIVE_FRAMES", "retained_native_events_per_orientation": 5056, + "retained_child_recipes": len(used), "endpoint_adapter_recipes": len(endpoints), + "residual_adapter_recipes": len(used_residuals), "new_input_recipes": len(records), + "new_exact_gaussian_columns": columns, "inverse_input_child_controls": len(records), + "phase_only_controls": len(records), "address_permutation_factors": 2 * len(records), + "method": "Independent signed Pauli and exact Gaussian interpretation; no compiler import", + "scope": "Positive inherited input frames before every retained adapter. No recursive hosting claim."} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/verify.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/verify.py new file mode 100644 index 0000000000..82a6d51d17 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/local/verify.py @@ -0,0 +1,137 @@ +"""Independent read-only exact local admission of a concrete borrowing variant.""" +import argparse +import copy +import gzip +import hashlib +import json +import sys +from pathlib import Path +from fractions import Fraction + +HERE=Path(__file__).resolve().parent +sys.path.insert(0,str(HERE)) +import native as C +from check_trace import run as replay +from precision import calculate + +def need(condition,label): + if not condition:raise ValueError(label) + +def rejected(function,label): + try:function() + except (ValueError,KeyError,AssertionError,StopIteration):return label + raise ValueError('mutation was accepted: '+label) + +def check_native(trace,native): + jobs,events,residuals=C.make_jobs(trace,literal_bound=True) + need(set(native['records'])==set(jobs),'complete native operation coverage') + need(native['event_recipe_refs']==events,'actual chronological native occurrence binding') + need(native['helper_residual_refs']==residuals,'actual relative helper binding') + for key,job in jobs.items(): + record=native['records'][key] + need(record['kind']==job['kind'],'native instruction kind') + need(json.loads(json.dumps(job['metadata']))==record['metadata'],'native frame/symbol metadata') + need(json.loads(json.dumps(job['actual'].circuit))==record['literal_expected'],'actual physical expected circuit') + need(record['recipe']['rank']==job['rank'],'actual paid Fourier rank') + C.N.equal(C.N.recipe(record['recipe']),tuple(job['actual'].circuit),'fresh independent native replay') + return jobs + +def check_wrappers(trace,native,wrappers): + helpers=wrappers['helpers'] + need(set(helpers)==set(native['helper_residual_refs']),'complete relative helper wrapper coverage') + for register,w in helpers.items(): + key=native['helper_residual_refs'][register];record=native['records'][key] + recipe=record['recipe'];endpoint=trace['endpoints'][register] + need(w['relative_native_ref']==key and w['width']==recipe['rank'],'actual helper wrapper binding') + need(w['initial']==endpoint['initial'] and w['final']==endpoint['final'],'actual nonzero helper entry/exit frames') + pre=C.N.adjoint(C.N.monomial(recipe['right']));post=C.N.adjoint(C.N.monomial(recipe['left'])) + need(json.loads(json.dumps(pre))==w['prewrapper'] and json.loads(json.dumps(post))==w['postwrapper'],'literal inverse monomial wrapper factors') + operation=C.chart(C.canonical(endpoint['initial'])).adjoint().then(C.chart(C.canonical(endpoint['final']))) + C.N.equal(pre+tuple(operation.circuit)+post,C.N.fourier(recipe['rank']),'normalized complete helper spectators and phase') + data=wrappers['data'];need(len(data)==224 and {e['register'] for e in data}==set(range(224)),'complete intrinsic data adapter coverage') + for e in data: + endpoint=trace['endpoints'][str(e['register'])] + source=(e['register']<112) if trace.get('orientation')!='backward' else (112<=e['register']<224) + if source: + old=C.canonical(endpoint['initial']);need(len(old)==1,'source singleton') + p=old[0];operation=C.CE.cgate(9,p).adjoint().then(C.chart(old)) + need(e['role']=='source' and e['position']=='entry','source adapter placement') + else: + new=C.canonical(endpoint['final']);normal=C.canonical(C.CE.nullspace(new,9));need(len(normal)==1,'target hyperplane') + p=normal[0];intrinsic=C.CE.cgate(9,p,-1).then(C.CE.fourier(9)) + operation=C.chart(new).adjoint().then(intrinsic) + need(e['role']=='target' and e['position']=='exit','target adapter placement') + need(e['normal']==p and e['recipe']['rank']==0,'actual intrinsic point and monomial rank') + C.N.equal(C.N.recipe(e['recipe']),tuple(operation.circuit),'intrinsic data adapter exact phase') + return dict(helper_wrappers=len(helpers),intrinsic_data_adapters=len(data)) + +def main(): + parser=argparse.ArgumentParser();parser.add_argument('--data',dest='variant',type=Path,default=HERE/'data');parser.add_argument('--out',type=Path) + args=parser.parse_args();reports={};hashes={};controls=[];postlude_reports={} + package=HERE.parent.resolve() + if args.out: + target=args.out.resolve() + need(target!=package and package not in target.parents,'receipt output must be outside the read-only package') + for pin in json.loads((HERE/'SOURCE-PINS.json').read_bytes())['files']: + need(hashlib.sha256((HERE/pin['path']).read_bytes()).hexdigest()==pin['sha256'],'portable source drift: '+pin['path']) + postludes=json.loads((args.variant/'postludes.json').read_bytes()) + for orientation in ('forward','backward'): + trace_path=args.variant/(orientation+'.json') + native_path=args.variant/('native-'+orientation+'.json.gz') + trace=json.loads(trace_path.read_bytes());native=json.loads(gzip.decompress(native_path.read_bytes())) + report=replay(trace);jobs=check_native(trace,native) + wrapper_path=args.variant/('wrappers-'+orientation+'.json') + wrappers=json.loads(wrapper_path.read_bytes()) + need(wrappers['trace_sha256']==hashlib.sha256(trace_path.read_bytes()).hexdigest(),'wrapper exact trace source binding') + need(wrappers['native_payload_sha256']==hashlib.sha256(gzip.decompress(native_path.read_bytes())).hexdigest(),'wrapper exact native payload binding') + report['wrappers']=check_wrappers(trace,native,wrappers) + report['native_jobs']=len(jobs);report['precision']=calculate(trace) + reports[orientation]=report + postlude=postludes[orientation] + private={e[1] for e in trace['events'] if e[0] in ('copy','zero_buffer')} + need(len(postlude)==len(private)==76 and {e[1] for e in postlude}==private,'exact complete owned-array CLEAR postlude') + need(all(e[0]=='clear' for e in postlude),'postlude has only actual exact CLEAR instructions') + composed=copy.deepcopy(trace);composed['events'].extend(postlude) + composed_report=replay(composed) + need(composed_report['paid_profile']==report['paid_profile'],'CLEAR postlude adds no recursive child') + need(composed_report['private_fields']==76,'postlude preserves actual private stock') + postlude_reports[orientation]=dict(exact_clear_instructions=len(postlude), + complete_public_columns=composed_report['independent_input_columns'], + every_owned_private_array_returns_zero=True,extra_paid_children=0, + actual_composed_word_sha256=hashlib.sha256(json.dumps(composed['events'],separators=(',',':')).encode()).hexdigest()) + hashes[orientation+'_trace']=hashlib.sha256(trace_path.read_bytes()).hexdigest() + hashes[orientation+'_native_payload']=hashlib.sha256(gzip.decompress(native_path.read_bytes())).hexdigest() + for kind in ('transport','helper_residual'): + key=next(k for k,v in jobs.items() if v['kind']==kind) + bad=copy.deepcopy(native['records'][key]['recipe']) + bad['left']['phase']=(bad['left']['phase']+1)%4 + controls.append(rejected(lambda:C.N.equal(C.N.recipe(bad),tuple(jobs[key]['actual'].circuit),'mutated phase'),orientation+'_'+kind+'_phase')) + if orientation=='forward': + bad=copy.deepcopy(trace) + event=next(e for e in reversed(bad['events']) if e[0]=='add' and 112<=e[1]<224) + event[3]=str(-Fraction(event[3])) + controls.append(rejected(lambda:replay(bad),'wrong_last_target_coefficient')) + bad=copy.deepcopy(trace) + index=next(i for i,e in enumerate(bad['events']) if e[0]=='handoff') + del bad['events'][index] + controls.append(rejected(lambda:replay(bad),'missing_actual_reference_change')) + bad=copy.deepcopy(native) + index=next(i for i,r in enumerate(bad['event_recipe_refs']) if r) + bad['event_recipe_refs'][index]=None + controls.append(rejected(lambda:need(bad['event_recipe_refs']==C.make_jobs(trace,literal_bound=True)[1],'binding changed'),'missing_native_occurrence_binding')) + print('PASS_INDEPENDENT_'+orientation.upper()+'_LOCAL_WORD',len(jobs),'native jobs',flush=True) + result=dict(status='PASS_COMPLETE_TWO_ORIENTATION_LOCAL_NATIVE_VARIANT', + reports=reports,source_hashes=hashes,negative_controls=controls, + exact_owned_clear_postludes=postlude_reports, + source_compiler_sha256=hashlib.sha256((C.VENDOR/'clifford_exact.py').read_bytes()).hexdigest(), + independent_native_interpreter_sha256=hashlib.sha256((C.VENDOR/'native_verify.py').read_bytes()).hexdigest(), + scope='Exact finite local words, arbitrary dirty logical restoration, owned-zero scratch and all actual native phase bindings. Whole-call owners, physical finite invoice, exact completed-child depth/precision admission, tape and all-size transfer remain separate.') + expected=json.loads((args.variant/'expected-local.json').read_bytes()) + for key in ('reports','source_hashes','negative_controls','source_compiler_sha256','independent_native_interpreter_sha256'): + need(json.loads(json.dumps(result[key]))==expected[key],'canonical exact local receipt mismatch: '+key) + if args.out: + args.out.parent.mkdir(parents=True,exist_ok=True) + args.out.write_text(json.dumps(result,indent=2)+'\n',encoding='utf-8') + print('PASS_PORTABLE_LOCAL_NATIVE_AND_EXACT_PRIVATE_POSTLUDE',len(controls),'fault controls',flush=True) + +if __name__=='__main__':main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/CHARTS.json.gz b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/CHARTS.json.gz new file mode 100644 index 0000000000..3f41867c60 Binary files /dev/null and b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/CHARTS.json.gz differ diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/EXPECTED.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/EXPECTED.json new file mode 100644 index 0000000000..4997a677ba --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/EXPECTED.json @@ -0,0 +1,49 @@ +{ + "status": "PASS_INDEPENDENT_ACTUAL_OWNER_CHARTS", + "roles": 31392, + "helpers_per_call": 436, + "actual_source_data_families_per_call": 224, + "actual_73column_charts": 7150, + "relative_endpoint_types": 26, + "entering_helper_roles": 14112, + "cross_replica_roles": 25555, + "derived_windows": 160, + "chart_digest": "19d3fd513eed08d51bbc348b9c83ed7806d551102b4288bdaec7a72a183e5abe", + "idle": { + "1": 1724, + "3": 20, + "5": 1469, + "6": 18, + "7": 494, + "32": 11, + "39": 7, + "48": 2493 + }, + "profile": { + "1": 176252, + "2": 30096, + "3": 30908, + "4": 10152, + "5": 4493, + "6": 18, + "7": 494, + "8": 576, + "32": 11, + "39": 7, + "48": 2493 + }, + "paid_mass": 520704, + "deficit": 2560, + "owners_sha256": "892b49f97d4caa7e92918262ea3a8cfa8cbf273c92d574ae53bb89a61ee5e847", + "source_trace_sha256": "a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee", + "parent_template_sha256": "f713f8e66aaeaebb8202fe72bf36c7d23497152497342c27892e0fda3092f95b", + "source_plan_sha256": "7b3122026e82b9d9e964d2a50c7bffdc976ae3dfea64bb98b2fdad5cfbdbaa20", + "actual_DATA_PHASES_sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa", + "controls_rejected": [ + "wrong-copy-index", + "wrong-family", + "overflow-offset", + "wrong-width", + "omitted-role" + ] +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/NOTICE.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/NOTICE.md new file mode 100644 index 0000000000..83a204ca95 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/NOTICE.md @@ -0,0 +1,19 @@ +# Ownership certificate attribution + +The new contribution is the selected 72-call whole-family owner assignment, +canonical physical-family metadata, source-bound touch-window reconstruction, +explicit 73-column orthogonal charts and independent read-only verifier. +This contribution was developed with OpenAI GPT-6 Astra and Codex assistance. + +The inherited Gauge44 parent and PLAN are retained byte-identically from +[PR1005](https://github.com/CrocSwap/integer-mult-bounds/pull/1005), credited to +icekylinx with the ancestry and AI disclosures preserved in +`notices/PR1005-NOTICE.md` and `notices/PR1005-UPSTREAM-NOTICE.md`. +The exact DATA-PHASES is the source pinned by that parent and its upstream pins; +the selected original local word is retained by the local package. Source paths, +commit references, and hashes appear in `SOURCES.json`. + +Apache-2.0 applies. The complete inherited license is `notices/LICENSE`. +No upstream credit, license, source pin or conditional scope is replaced by this +notice. The finite certificate and its replay do not claim to discharge the +retained all-size multiplication interfaces. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/OWNERS.json.gz b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/OWNERS.json.gz new file mode 100644 index 0000000000..912df33d6e Binary files /dev/null and b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/OWNERS.json.gz differ diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/SOURCES.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/SOURCES.json new file mode 100644 index 0000000000..9d64a09e82 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/SOURCES.json @@ -0,0 +1,74 @@ +{ + "schema": "selective-borrowing-owner-sources/1", + "inputs": [ + { + "path": "OWNERS.json.gz", + "sha256": "892b49f97d4caa7e92918262ea3a8cfa8cbf273c92d574ae53bb89a61ee5e847" + }, + { + "path": "CHARTS.json.gz", + "sha256": "a7971424df309a4191754bb001cea442282f6e769385a5966bbd14d6071899ce" + }, + { + "path": "inputs/HOSTED-PARENT.json.gz", + "sha256": "f713f8e66aaeaebb8202fe72bf36c7d23497152497342c27892e0fda3092f95b" + }, + { + "path": "inputs/PLAN.json", + "sha256": "7b3122026e82b9d9e964d2a50c7bffdc976ae3dfea64bb98b2fdad5cfbdbaa20" + }, + { + "path": "inputs/DATA-PHASES.json", + "sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa" + }, + { + "path": "notices/LICENSE", + "sha256": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4" + }, + { + "path": "notices/PR1005-NOTICE.md", + "sha256": "7a8d17b04da1e6c9d0ec217fb5bda72faa09d9a21695128722dd1e2827870fcf" + }, + { + "path": "notices/PR1005-UPSTREAM-NOTICE.md", + "sha256": "a75967ebc34e3bf2ff2c25c08e6c7f1a61b0c0d171cfd1b62196ae2caaf2c610" + }, + { + "path": "../local/data/forward.json", + "sha256": "a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee" + }, + { + "path": "../local/inputs/selected-complex-word.json.gz", + "sha256": "16bc0ebb04b4be6c0ade74badc6a0c0cdba4215eda86e4ff9f13abf8ecebd219" + } + ], + "upstream": { + "pr1005": { + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1005", + "repository": "jeremykothe2/integer-mult-bounds", + "head": "779e80aef469bc3a28b19f1adde199871a9a22e3", + "retained_paths": { + "inputs/HOSTED-PARENT.json.gz": "research/gauge44-leaver8/evidence/global/GLOBAL-TEMPLATE.json.gz", + "inputs/PLAN.json": "research/gauge44-leaver8/data/PLAN.json", + "notices/LICENSE": "research/gauge44-leaver8/LICENSE", + "notices/PR1005-NOTICE.md": "research/gauge44-leaver8/NOTICE.md", + "notices/PR1005-UPSTREAM-NOTICE.md": "research/gauge44-leaver8/UPSTREAM-NOTICE.md" + } + }, + "data_phases": { + "path": "research/community-k492-audit/pr622/replay/baseline558/complex-global/base/work/complex-nine-native/DATA-PHASES.json", + "binding": "exact source hash retained in PR1005 UPSTREAM-PINS.json and HOSTED-PARENT exact_data_phase_recipe_sha256" + }, + "local_word": { + "base_commit": "697b00e5d36297563c22aac080dd5dee398c2624", + "path": "research/complex-block-rank-optimum/inputs/selected.json.gz", + "git_blob": "5f9ef680983b0b63a4c82a4ed909837b78e690cd", + "binding": "retained byte-identical through PR2226 input selected-complex-word.json.gz" + } + }, + "new_artifacts": { + "OWNERS.json.gz": "Actual 72-call whole-family assignment; copy ABI canonicalized before all final checks.", + "CHARTS.json.gz": "Independently reconstructed complete physical charts, routes, and nonzero idle suffixes. Empty-child translations remain in the complete inherited parent, not this rank-factor-only chart view." + }, + "source_policy": "Pinned upstream input bytes and notices are retained. The verifier regenerates the certificate from the actual parent execution schedule and actual DATA-PHASES, importing no search or optimizer module." +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/inputs/DATA-PHASES.json b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/inputs/DATA-PHASES.json new file mode 100644 index 0000000000..d9a4f84211 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/inputs/DATA-PHASES.json @@ -0,0 +1 @@ +{"cover_correction":{"complete_small_cover_checks":[{"n":1,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":1},{"n":2,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":2},{"n":3,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":6},{"n":4,"nonzero_full_kernel_corrections":24,"orthogonal_matrices":48},{"n":5,"nonzero_full_kernel_corrections":600,"orthogonal_matrices":720}],"formula":"delta(g)_i = (weight(row_i(g))-1)/2 mod 2; apply X_delta(g) to EVERY final live stream in cover class g.","mandatory":true,"reason":"P_g C_all P_g^-1 = C_all X_delta(g) exactly. Orthogonality cancels all quadratic cross terms; both sides fix the uniform vector."},"global_phase_argument":"Every C_q and every linear address permutation fixes the uniform vector. 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"c682e9ff1b347f89771c6800f46ad98c2bb92b91c5c88452d53a9e7f0180c35f" + } + } + }, + "source_plan_sha256": "258a8b9baa245fa4096828671e2f1d1462b5af08d56890629622ef6125eaddda", + "assignment_sha256": "32ca30cd1c652ef6ad43657273a08a157e1df3545e759ab36306a0d0b516fb4a", + "source_owners_sha256": "935ff1a6823c6625daba3349b28d74acf9a3d70a5947ae155aa66c7821804add" +} diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/notices/LICENSE b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/notices/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/notices/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. 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We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/notices/PR1005-NOTICE.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/notices/PR1005-NOTICE.md new file mode 100644 index 0000000000..30d0c8cd3c --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/notices/PR1005-NOTICE.md @@ -0,0 +1,24 @@ +# Gauge44 attribution and scope + +New contributions by icekylinx, with OpenAI GPT-6 Astra assistance. +Local review and packaging used OpenAI Codex. Apache-2.0 applies; +the license is included. The full original PR635 notice is preserved as +UPSTREAM-NOTICE.md and construction/UPSTREAM-NOTICE.md. + +The new contribution is the changed source-branch gauge and retired-row +completion, source-carried coordinate totals, exact 44-cold-end realization, +and explicit all72 whole-family assignment using concave capacity matching +with integer majorization certificates. The center-carrier generalization +retains the exact ordered common frames and complete native local contract. +Independent local, price, interface and owner fixed-point checks are included. + +The construction retains the credited PR635 K-triangle and porting machinery, +earlier joint-prefix/HPE/frame ancestry and icekylinx's prior factory work. +#645 used the same cited PR543 bit supplier as PR635. This package (research/gauge44-leaver8) cites the merged PR589 bit supplier instead, with a derived rank-mass deficit (PROOF-NOTE-BIT.md); #645's retained PR543-based evidence is historical. It builds #645's word without K-triangles (PartialK) and stacks earlier community jp8 layers on it (G44Stack), adds the p6n ends (NoTriReopen), entrance promotions (BlockInvert) a read-set exchange (BRoute), chafreaky's #741 frames, JointXF's joint exchange, NewBArch's triangle-hybrid B rewrite and LeaverSAT's eight k8 leaver cells (with LeaverSAT2's frame re-optimisation); see README. All-size, realization, +hosting, precision/tape, analytic-transfer and bit H1-H5 assumptions remain +conditional. No upstream license, credit, source pin or AI disclosure is +replaced by this notice. + +PR641 and PR642 were checked as concurrent reported frontier comparisons. +Their endpoints and owner maps are not included here. See CONSTRUCTION.md, +hosting/PROOF.md, independent/ and README.md for the exact contribution scope. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/notices/PR1005-UPSTREAM-NOTICE.md b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/notices/PR1005-UPSTREAM-NOTICE.md new file mode 100644 index 0000000000..ebc24a0168 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/notices/PR1005-UPSTREAM-NOTICE.md @@ -0,0 +1,34 @@ +# Attribution and scope + +Four verified provider types are offered to every call: PR571's prefix word with twelve multi-cut cold kernels (PR568's unchanged constructor), the same word with forty width-five entrance promotions (PR569's unchanged frame constructor) before those kernels, PR575's plain joint-prefix provider, and PR568's eight-cold provider. A window MILP with an exact per-pattern model chooses the dispatch and a literal owner map. PR583's six source-bound kernel frame coalescences (its unchanged constructor and negative controls) are applied to every PR571-prefix provider word (ck and atm3), followed by two further coalescences that each first move one addition past two commuting additions (package code/build_gate_moves.py, extending PR583's constructor); a final hyperplane_end step (package code/build_hpe.py) then runs 46 single-gate cleanups h -= p right after part B at the dim-8 join U of their registers, so 46 width-8 entering helpers per call end at U with width 7; each such gate commutes with every earlier cleanup gate on its registers (for 8 of them the partner's one earlier gate x -= p only reads p), and the reordered program is replayed exactly. A window MILP (exact windows, RINS around an LP vertex) gives the dispatch ck 45 / atm3 21 / jp8 6 and the literal owner map. + +Pinned external sources (unchanged; `verify.py` checks their manifests and every file it uses): +- PR568 (`research/complex-multicold`, commit `71c99ac935a33bb7b59a1b4a7af8725a939b0b90`). +- PR571 (`research/structural-gain`, commit `b7bdc5a8de18fcf418cfb9c7d83a2b1097a5e4a3`). +- PR569 (`research/larger-gain`, commit `897213a855b7fb8a1b8b0e552a2f5c42e6c1b326`). +- PR575 (`research/complex-joint-prefix`, commit `eac6f7e378ab622dac50a2fd4b7ef6eab014de44`). +- PR583 (`research/kernel-frame-coalescing`, commit `89be768f0157f5a3d8e4db69cf1748768605eac9`). + +Provider construction is described as typed steps in `data/SPEC.json` (`build`); `verify.py` runs only allowlisted step types with their paired omission audits and checks every input and output content hash. `code/` holds the spec-driven global admission scripts adapted from PR571's `frontier-composition/global-code/` (OpenAI Codex assisted, Apache-2.0): every expected value is read from `data/SPEC.json` and compared only after independent recomputation. `code/generated_identity.py` and `code/generated_identity_controls.py`, and the decompressed-content artifact identities in the bind, audit, invoice, assembly and phase-timing scripts, are taken unchanged from PR586 by chafreaky (`research/promoted-kernel-coalescing`, commit `4846fee5b06af770e9b3a10c36200f673d656725`, OpenAI Codex assisted, Apache-2.0), including its proof-content and reference/metadata mutation controls; the normalization of descriptive width metadata after frame changes follows the same PR. + +In PR601, the owner map was nominated by a HiGHS window MILP (adapted from PR569's `global/nominate_width7.py`) with lazy cross-time Hall cuts; this package's re-solved map is described below. Neither is an admission dependency. + +DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code. (The PR601 package this one starts from states: "Prepared by Claude Code (Anthropic) research agents.") Apache-2.0 applies; LICENSE is included. All notices, licenses, source pins and AI disclosures of the pinned sources and their ancestry are retained in those packages and are not replaced by this notice. All inherited conditional assumptions remain conditional. + +**This package (complex-571-cold14, DreamingOfClouds).** It starts from PR601 (jeremykothe2, `research/complex-571-hpem46`, commit `529ab7dcb59d5ea7a5798e9e06cf5b3aee8a1550`). Everything above is kept unchanged except the following: +- every provider chain (ck, atm3, jp8) gains the typed step `cold_end` (`code/build_cold_end.py`, `code/cold_end_controls.py`; recipes `data/recipes/*-cold_end.json.gz`); +- the generalised native chain and the global admission scripts accept cold helpers that end below FULL (with `0:U` wrappers), a refactored exact cleanup, and check the wrapper order; +- `data/PLAN.json` / `data/OWNERS.json.gz` (dispatch and literal owner map) come from a relocate/swap/ejection-chain/provider-flip annealing search started from PR601's map, refined by an exact pattern-restricted owner model (CP-SAT) with a literal edge-colouring expansion (not an admission dependency); +- `data/SPEC.json` holds the expected values, recorded by a discovery run and then checked by a strict run. + +The cold-end construction (a row-plus-column cone condition for low-frame ends, a rank-0 dirt-spread premix and multi-gate lower-frame cleanup) builds on PR591 / PR592 / PR593 / PR601 (jeremykothe2) and our PR599 (Lean: multi-gate and cold hyperplane-end tails). DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code. Build on this at your own risk. + +**This package (complex-571-cold14-b3, jeremykothe2).** It is PR604's `complex-571-cold14` (DreamingOfClouds) with ColdBeyond's second cold end per block (two new `cold_end` A-gate primitives, `PROOF-NOTE.md`), ColdSupply's E1 entering pair and reviewed controls, a free dispatch (jp8 ×60, atm3 ×12) and a literal owner map from an exact per-time-Hall window MIP. All other construction, code, controls and proofs are PR604's and earlier contributors', unchanged. + +**This package (complex-571-cold14-k492).** It is complex-571-cold14-b3 with these changes: +- **K-triangle base words.** The leaver-read K-triangle skeleton atom and its read schedule (`data/TRI-SCHEDULE.json`) are by AtomScout, a Claude Code research agent. `code/construct_k_triangle.py` is PR531's `code/construct_504.py` (icekylinx, OpenAI Codex / GPT assisted, Apache-2.0; pinned source `direct_delivery`) extended with that atom. +- **Ported chains.** The PR604/B3 chains are ported stage by stage onto the K-triangle words by `code/build_port_stage.py` (Port480, a Claude Code research agent). `code/build_frame_regroup.py` extends PR583's `build_frames.py` (chafreaky). +- **Helper count R.** The generalisation of the global scripts to the provider helper count R is by GlobalR (a Claude Code research agent), with AtomReviewer's fixes (PROOF-NOTE-R.md). +- **Owner map.** The owner map, the cold_end recipes on the K-triangle words and the package assembly are by Stack492, Port480 and ColdBeyond (Claude Code research agents). + +Build on this at your own risk. diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/verify.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/verify.py new file mode 100644 index 0000000000..3b6b9b0a3b --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/owners/verify.py @@ -0,0 +1,154 @@ +"""Independent full literal ownership and physical orthogonal chart reconstruction. + +Consumes source DATA-PHASES, actual serialized parent schedule and fixed local +endpoint trace. Imports no emitter, optimizer, flow or upstream audit module. +""" +from collections import Counter,defaultdict +from pathlib import Path +import argparse,gzip,hashlib,json,copy + +HERE=Path(__file__).resolve().parent +PACKAGE=HERE.parent +BASE=HERE/'inputs' +PHASE=BASE/'DATA-PHASES.json' +WORD=PACKAGE/'local/inputs/selected-complex-word.json.gz' +PINS={ + 'OWNERS.json.gz':'892b49f97d4caa7e92918262ea3a8cfa8cbf273c92d574ae53bb89a61ee5e847', + 'inputs/HOSTED-PARENT.json.gz':'f713f8e66aaeaebb8202fe72bf36c7d23497152497342c27892e0fda3092f95b', + 'inputs/PLAN.json':'7b3122026e82b9d9e964d2a50c7bffdc976ae3dfea64bb98b2fdad5cfbdbaa20', + 'inputs/DATA-PHASES.json':'c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa', + '../local/data/forward.json':'a9d513bc1d85eb11184d73242a4016119f3968bb2a1ff9b365e78109778c17ee', + '../local/inputs/selected-complex-word.json.gz':'16bc0ebb04b4be6c0ade74badc6a0c0cdba4215eda86e4ff9f13abf8ecebd219', +} + +def need(ok,msg): + if not ok:raise ValueError(msg) +def read(p):return json.loads(gzip.decompress(p.read_bytes())if p.suffix=='.gz'else p.read_bytes()) +def sha(p):return hashlib.sha256(p.read_bytes()).hexdigest() +def span(xs): + out={0} + for x in xs:out|={z^x for z in list(out)} + return out +def basis(q): + zero=next(i for i in range(9)if not(q>>i&1));a=q^(1<>i&1 else 0)for i in order]+[1<=224}),entering_helper_roles=enters,cross_replica_roles=cross,derived_windows=len(c['windows']),chart_digest=digest.hexdigest(),idle=dict(sorted(idle.items())),profile=dict(sorted(profile.items())),paid_mass=mass,deficit=2560) + return result,dict(schema='actual-whole-owner-orthogonal-charts/1',unique_73column_charts=sigmas,whole_helper_routes=routes,actual_idle_complements=idle_rows) + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out',type=Path,help='Optional external directory for regenerated receipt and chart certificate') + args=parser.parse_args() + if args.out is not None: + dest=args.out.resolve() + need(not dest.is_relative_to(PACKAGE.resolve()),'--out must be outside the published package') + need(not (dest/'OWNERS-RECEIPT.json').exists() and not (dest/'OWNERS-CHARTS.json.gz').exists(),'refuse to overwrite an existing replay receipt') + for rel,expected in PINS.items(): + need(sha(HERE/rel)==expected,'immutable source pin: '+rel) + owners=HERE/'OWNERS.json.gz' + trace=PACKAGE/'local/data/forward.json' + c=context(trace) + rows=read(owners) + result,charts=check(rows,c,True) + need(charts==read(HERE/'CHARTS.json.gz'),'complete regenerated physical chart certificate') + controls=[] + for name in ('wrong-copy-index','wrong-family','overflow-offset','wrong-width','omitted-role'): + bad=copy.deepcopy(rows) + if name=='wrong-copy-index':bad[0]['copy']+=112 + elif name=='wrong-family':bad[0]['family']+=1 + elif name=='overflow-offset':bad[0]['offset']=73 + elif name=='wrong-width':bad[0]['width']+=1 + else:bad.pop() + try:check(bad,c) + except (ValueError,KeyError):controls.append(name) + else:raise ValueError('owner mutation accepted: '+name) + result.update(owners_sha256=sha(owners),source_trace_sha256=sha(trace),parent_template_sha256=sha(c['parent_path']),source_plan_sha256=sha(BASE/'PLAN.json'),actual_DATA_PHASES_sha256=sha(PHASE),controls_rejected=controls) + # JSON round-trip normalizes integer rank keys to the public JSON ABI. + result=json.loads(json.dumps(result)) + need(result==read(HERE/'EXPECTED.json'),'complete expected mathematical receipt') + if args.out is not None: + dest.mkdir(parents=True,exist_ok=True) + (dest/'OWNERS-RECEIPT.json').write_text(json.dumps(result,indent=2)+'\n',encoding='utf-8') + (dest/'OWNERS-CHARTS.json.gz').write_bytes(gzip.compress(json.dumps(charts,sort_keys=True,separators=(',',':')).encode(),mtime=0)) + print(json.dumps(result,indent=2)) + +if __name__=='__main__':main() diff --git a/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/verify.py b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/verify.py new file mode 100644 index 0000000000..a384dcbc48 --- /dev/null +++ b/research/selective-borrowing-outer-retune/vendor/pr2260-selective-borrowing-kappa/verify.py @@ -0,0 +1,43 @@ +#!/usr/bin/env python3 +"""Fresh, read-only complete finite conditional supplier replay (stdlib only).""" +import argparse,gzip,hashlib,json,subprocess,sys,tempfile,time +from pathlib import Path +sys.dont_write_bytecode=True +HERE=Path(__file__).resolve().parent + +def need(ok,label): + if not ok:raise ValueError(label) +def read(path):return json.loads(path.read_bytes()) +def snapshot(): + return {p.relative_to(HERE).as_posix():hashlib.sha256(p.read_bytes()).hexdigest() for p in HERE.rglob('*') if p.is_file()} +def run(work): + before=snapshot();manifest=read(HERE/'MANIFEST.json')['sha256'] + need(set(before)==set(manifest)|{'MANIFEST.json'},'package contains missing or undeclared files') + need(all(before[k]==v for k,v in manifest.items()),'frozen package SHA256 differs') + def command(script,*args): + subprocess.run([sys.executable,'-I','-O','-B',str(HERE/script),*map(str,args)],check=True,timeout=500) + command('local/verify.py','--out',work/'LOCAL.json') + command('owners/verify.py','--out',work/'owners') + command('independent/verify.py','--work',work/'independent') + command('local/invoice.py','--variant',HERE/'local/data','--owners',HERE/'owners/OWNERS.json.gz','--out',work/'INVOICE.json') + need(read(work/'INVOICE.json')==read(HERE/'local/data/finite-invoice.json'),'fresh finite invoice differs') + command('arithmetic/verify.py','--out',work/'ARITHMETIC.json') + local=read(work/'LOCAL.json');owner=read(work/'owners/OWNERS-RECEIPT.json');invoice=read(work/'INVOICE.json');arithmetic=read(work/'ARITHMETIC.json') + need(local['source_hashes']['forward_trace']==owner['source_trace_sha256'],'local and owner word differ') + need(owner['owners_sha256']==invoice['owner_table_sha256'],'owner and finite invoice differ') + need(arithmetic['source_sha256']['owners/OWNERS.json.gz']==owner['owners_sha256'],'arithmetic owner differs') + need(arithmetic['profile']==owner['profile'],'complete paid profiles differ') + need(arithmetic['kappa']=='3361/2000000','wrong final kappa') + need(snapshot()==before,'source package changed during replay') + result=dict(status='PASS_COMPLETE_FINITE_CONDITIONAL_KAPPA',kappa=arithmetic['kappa'],kappa_decimal=arithmetic['kappa_decimal'],source_manifest_sha256=hashlib.sha256((HERE/'MANIFEST.json').read_bytes()).hexdigest(),all_source_files_unchanged=True,physical_roles=owner['roles'],physical_stock=invoice['physical_families'],native_jobs=sum(r['native_jobs'] for r in local['reports'].values()),strict_outer_rows=len(arithmetic['outer_constraints']),scope='All supplied finite identities, physical factor partition, exact fine-grid invoice and rational asymptotic assembly checked. Inherits original all-size compiler/physical fixed-tape/analytic transfer and retained bit supplier assumptions. Optional Lean lemmas are separate; this replay is not a Lean proof of multiplication.') + (work/'RESULT.json').write_text(json.dumps(result,indent=2)+'\n') + print(json.dumps(result,indent=2),flush=True) + +def main(): + parser=argparse.ArgumentParser(description=__doc__);parser.add_argument('--out',type=Path,help='Nonexistent external directory for receipts (otherwise temporary).');args=parser.parse_args() + if args.out: + need(not args.out.resolve().is_relative_to(HERE),'output must be outside source package') + args.out.mkdir(parents=True,exist_ok=False);run(args.out) + else: + with tempfile.TemporaryDirectory(prefix='selective-borrowing-kappa-') as d:run(Path(d)) +if __name__=='__main__':main() diff --git a/research/selective-borrowing-outer-retune/verify.py b/research/selective-borrowing-outer-retune/verify.py new file mode 100644 index 0000000000..5eb7187c72 --- /dev/null +++ b/research/selective-borrowing-outer-retune/verify.py @@ -0,0 +1,93 @@ +#!/usr/bin/env python3 +"""Verify from scratch: PR2260's supplier (vendored at its head commit) + certified fine endpoint + PR1673 capped phase-cell +fusion outer with retuned strict slacks (the PR2220 retune method). + +Steps + 1. MANIFEST.json: every package file's SHA-256. + 2. Supplier replay = the steps of vendor/pr2260-selective-borrowing-kappa/verify.py (local natives, owners, independent + family audit, finite invoice, exact arithmetic incl. PR2260's own 47-row outer), run unmodified from the vendored copy. + One deviation, stated: PR2260's independent receipt pins the SHA-256 of a gzip.compress(...) byte stream, which depends on + the zlib build; here every other receipt key must equal PR2260's EXPECTED.json and the UNCOMPRESSED payload is pinned + (SUPPLIER-BINDING.json). + 3. Binding: PR2260's recounted global profile, live stock and bit endpoint equal this package's SPEC.json. + 4. certify_endpoint.py: complex endpoint C to 10^-200 (native and conservative moments < 1 at C; conservative > 1 at C+10^-200). + 5. outer/run_outer.py: regenerates LEDGER.json / INDEPENDENT.json byte-identically and replays the supplier-independent + PR1673 / PR2220 outer evidence (needs numpy and SymPy, as PR2220). +usage: python3 -B verify.py [--skip-supplier] (assertions must be enabled) +""" +import gzip, hashlib, json, subprocess, sys, tempfile +from fractions import Fraction as Q +from pathlib import Path +sys.set_int_max_str_digits(0) +if not __debug__: raise RuntimeError('Assertions must be enabled') +HERE = Path(__file__).resolve().parent +SUP = HERE / 'vendor/pr2260-selective-borrowing-kappa' +def need(ok, why): + if not ok: raise SystemExit('FAIL: ' + why) +def sha(p): return hashlib.sha256(Path(p).read_bytes()).hexdigest() +def rd(p): return json.loads(Path(p).read_bytes()) + +man = rd(HERE / 'MANIFEST.json')['sha256'] +files = {p.relative_to(HERE).as_posix() for p in HERE.rglob('*') if p.is_file() and '__pycache__' not in p.parts} +need(files == set(man) | {'MANIFEST.json'}, 'package contains missing or undeclared files') +need(all(sha(HERE / k) == v for k, v in man.items()), 'package file SHA-256 differs') +print('PASS_MANIFEST', len(man)) + +spec = rd(HERE / 'SPEC.json'); bind = rd(HERE / 'SUPPLIER-BINDING.json') +need(sha(SUP / 'MANIFEST.json') == bind['pr2260_manifest_sha256'], 'vendored PR2260 manifest differs') +with tempfile.TemporaryDirectory(prefix='pr2260-retune-') as d: + work = Path(d) + if '--skip-supplier' not in sys.argv[1:]: + def command(script, *args): + subprocess.run([sys.executable, '-I', '-O', '-B', str(SUP / script), *map(str, args)], check=True, timeout=900) + command('local/verify.py', '--out', work / 'LOCAL.json') + command('owners/verify.py', '--out', work / 'owners') + # PR2260 independent family audit, run through its own audit.audit with its own arguments + sys.path.insert(0, str(SUP / 'independent')) + import importlib.util, types + for rel, pin in rd(SUP / 'independent/SOURCES.json').items(): + need(sha(SUP / rel) == pin, 'PR2260 independent source differs: ' + rel) + spec_a = importlib.util.spec_from_file_location('pr2260_independent_audit', SUP / 'independent/audit.py') + audit = importlib.util.module_from_spec(spec_a); sys.modules[spec_a.name] = audit; spec_a.loader.exec_module(audit) + R = SUP + args = types.SimpleNamespace(parent=R / 'independent/inputs/T8-PARENT.json.gz', + parent_sha256='025448abdf2012d327e50a009084e1501690d05e650300d2982836aa0350ae00', + phases=R / 'owners/inputs/DATA-PHASES.json', plan=R / 'owners/inputs/PLAN.json', owners=R / 'owners/OWNERS.json.gz', + forward_trace=R / 'local/data/forward.json', backward_trace=R / 'local/data/backward.json', + forward_native=R / 'local/data/native-forward.json.gz', backward_native=R / 'local/data/native-backward.json.gz', + forward_wrapper=R / 'local/data/wrappers-forward.json', backward_wrapper=R / 'local/data/wrappers-backward.json', + engine=R / 'local/vendor/native_verify.py', out=work / 'independent') + audit.audit(args) + actual = rd(work / 'independent/RESULT.json'); expected = rd(SUP / 'independent/EXPECTED.json') + for k, v in expected.items(): + if k == 'actual_factors_sha256': continue + need(actual.get(k) == v, 'PR2260 independent receipt differs: ' + k) + payload = hashlib.sha256(gzip.decompress((work / 'independent/ACTUAL-FAMILY-FACTORS.json.gz').read_bytes())).hexdigest() + need(payload == bind['independent_actual_factors_uncompressed_sha256'], 'PR2260 family-factor payload differs') + print('PASS_PR2260_INDEPENDENT (all receipt keys; uncompressed factor payload pinned)') + command('local/invoice.py', '--variant', SUP / 'local/data', '--owners', SUP / 'owners/OWNERS.json.gz', '--out', work / 'INVOICE.json') + need(rd(work / 'INVOICE.json') == rd(SUP / 'local/data/finite-invoice.json'), 'fresh finite invoice differs') + command('arithmetic/verify.py', '--out', work / 'ARITHMETIC.json') + local = rd(work / 'LOCAL.json'); owner = rd(work / 'owners/OWNERS-RECEIPT.json'); inv = rd(work / 'INVOICE.json'); ar = rd(work / 'ARITHMETIC.json') + need(local['source_hashes']['forward_trace'] == owner['source_trace_sha256'], 'local and owner word differ') + need(owner['owners_sha256'] == inv['owner_table_sha256'], 'owner and finite invoice differ') + need(ar['source_sha256']['owners/OWNERS.json.gz'] == owner['owners_sha256'], 'arithmetic owner differs') + need(ar['profile'] == owner['profile'] == actual['complete_profile'], 'complete paid profiles differ') + need(ar['kappa'] == '3361/2000000', 'PR2260 supplier kappa differs') + # 3. binding to this package + need({str(k): v for k, v in ar['profile'].items()} == spec['expected']['global_profile'], 'SPEC profile is not PR2260 recount') + need(Q(ar['inherited_bit_saving']) == Q(spec['supplier']['bit_lower']), 'bit endpoint differs') + print('PASS_PR2260_SUPPLIER_REPLAY_AND_BINDING') + else: + print('SKIPPED supplier replay (--skip-supplier)') + need(rd(SUP / 'arithmetic/EXPECTED.json')['profile'] == spec['expected']['global_profile'], 'SPEC profile differs from PR2260 EXPECTED') + subprocess.run([sys.executable, '-B', str(HERE / 'certify_endpoint.py')], check=True) + subprocess.run([sys.executable, '-B', str(HERE / 'outer/run_outer.py'), str(work / 'outer')], check=True) +led = rd(HERE / 'outer/LEDGER.json') +row = led['rows']['struct_boundary_radix_union'] +k = row['kappa']['decimal'] +kq = Q(row['kappa']['rational']) +need(kq == Q(rd(HERE / 'ENDPOINT.json')['complex_lower']) - Q(1, 10 ** 246) - Q(1, 10 ** 222), 'kappa is not C - 1/F - B') +print('PASS_COMPLETE kappa =', k[:60], '...') +print(' over PR2260 (3361/2000000): +%.6f%%' % float((kq / Q(3361, 2000000) - 1) * 100)) +print(' over PR2220: +%.6f%%' % float((kq / Q('0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263') - 1) * 100))